TORSION UNITS OF INTEGRAL GROUP RINGS AND SCHEME RINGS

A Thesis

Submitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirements

For the Degree of

Doctor of Philosophy

In

Mathematics

University of Regina

By

Gurmail Singh

Regina, Saskatchewan

August, 2015

c© Copyright 2015: Gurmail Singh

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Gurmail Singh, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Torsion Units of Integral Group Rings and Scheme Rings, in an oral examination held on August 26, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: *Dr. Yuanlin Li, Brock University

Co-Supervisor: Dr. Allen Herman, Department of Mathematics & Statistics

Co-Supervisor: Dr. Shaun Fallat, Department of Mathematics & Statistics

Committee Member: Dr. Fernando Szechtman, Department of Mathematics & Statistics

Committee Member: Dr. Karen Meagher, Department of Mathematics & Statistics

Committee Member: **Dr. Robert Hilderman, Department of Computer Science

Chair of Defense: Dr. Christopher Yost, Department of Biology *via SKYPE **Not present at defense

Abstract

We study torsion units of algebras over the ring of integers Z with nice bases.

These include integral group rings, integral adjacency algebras of association

schemes and integral C-algebras.

Torsion units of group rings have been studied extensively since the 1960’s.

Much of the attention has been devoted to the Zassenhaus conjecture for normal-

ized torsion units of ZG, which says that they should be rationally conjugate (i.e.

inQG) to elements of the group G. In recent years several new restrictions on inte-

gral partial augmentations for torsion units of ZG have been introduced that have

improved the effectiveness of the Luthar-Passi method for checking the Zassen-

haus conjecture for specific finite groups G. We have implemented a computer

program that constructs units of QG that have integral partial augmentations that

are relevant to the Zassenhaus conjecture. Indeed, any unit of ZG with these par-

tial augmentations would be a counterexample to the conjecture. In all but three

i

exceptions among groups of order less than 160, we have constructed units of

QG with these partial augmentations that satisfy a condition which implies they

cannot be rationally conjugate to an element of ZG. Currently our package has

computational difficulties with the Luthar-Passi method for some of the groups of

order 160.

As C-algebras are generalization of groups, it is natural to ask about torsion

units of C-algebras. We establish some basic results about torsion units of C-

algebras analogous to what happens for torsion units of group rings. These results

can be immediately applied to give new results for Schur rings, Hecke algebras,

adjacency algebras of association schemes and fusion rings. We also investigate

the possibility for a conjecture analogous to the Zassenhaus conjecture in the C-

algebra setting.

ii

Acknowledgment

I am grateful to my supervisors Professor Allen W. Herman and Professor Shaun

M. Fallat for all of the their support, understanding, patience, and knowledge they

have provided me over the course of my study. Without their supervision and

mentorship this would not have been possible.

I am thankful to Dr. Yuanlin Li, my external examiner. I also wish to thank

Dr. Karen Meagher and Dr. Fernando Szechtman for their advice and careful

reading of the manuscript. Their suggestions have been valuable and helpful for

my thesis.

Finally, the financial support of my supervisors’s NSERC grants, Department

of Mathematics and Statistics, and the Faculty of Graduate Studies and Research

during my PhD program allowed me to focus solely on my PhD program during

these past years.

iii

To my family

iv

Contents

Acknowledgment iii

Dedication iv

1 Introduction 1

2 Background 5

2.1 Group rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Generalized C-algebras and table algebras . . . . . . . . . . . . . 9

2.3 Association schemes and scheme rings . . . . . . . . . . . . . . . 12

2.4 Representations and characters of semisimple algebras . . . . . . 18

2.4.1 Semisimple algebras . . . . . . . . . . . . . . . . . . . . 18

2.4.2 Representation theory of semisimple algebras . . . . . . . 21

2.4.3 Representation theory of groups . . . . . . . . . . . . . . 24

v

3 Basic Tools 28

3.1 Torsion units of ZG and partial augmentations . . . . . . . . . . . 29

3.2 Luthar-Passi method . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 The standard feasible trace of a C-algebra . . . . . . . . . . . . . 36

4 Normalized Torsion Units of Integral Group Rings 45

4.1 Partial augmentations and rational conjugacy . . . . . . . . . . . 46

4.2 Computer implementation of the Luthar-Passi method . . . . . . . 49

4.3 Computer construction of torsion units with prescribed partial aug-

mentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Partially central torsion units of QG . . . . . . . . . . . . . . . . 65

5 Torsion units of C-algebras 69

5.1 Torsion units of RB . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Torsion units for integral C-algebras with a

standard character . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Lagrange’s theorem for normalized torsion

units of ZB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Applications to Schur rings and Hecke algebras . . . . . . . . . . 90

5.4.1 Schur rings . . . . . . . . . . . . . . . . . . . . . . . . . 90

vi

5.4.2 Integral Hecke algebras . . . . . . . . . . . . . . . . . . . 92

6 Future Work 93

6.1 Categorical aspects . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 When all units of ZB are trivial . . . . . . . . . . . . . . . . . . . 96

6.3 Normalized automorphisms of QB . . . . . . . . . . . . . . . . . 98

vii

Chapter 1

Introduction

Let R be a commutative ring with identity. A group ring RG is a ring as well

as a free R-module whose basis is a multiplicative group G. When the ring R is

replaced with a field K then KG is called a group algebra. The trivial units of a

group ring RG are scalar multiples of a single group element in G by a unit of

the ring R. We will investigate a conjecture that would characterize the nontrivial

torsion units of the integral group ring ZG. To get a broader perspective of what

is really going on, we formulate and study the analogous conjecture in the general

settings of the integral C-algebras and integral adjacency algebras of association

schemes (a.k.a scheme rings). Along the way we establish several basic results to

set up the necessary machinery in these new settings.

1

In this dissertation we present new results on torsion units of group rings,

C-algebras, and scheme rings. Using the Luthar-Passi method and our new con-

struction for partially central units we show how we have verified the Zassenhaus

conjecture for all but three groups of order up to 159. In the case of C-algebras we

establish some fundamental results for torsion units of integral C-algebras with s-

tandard character, and we prove a Lagrange-type theorem for integral C-algebras

that applies directly to integral scheme rings.

Our initial motivation came from the fact that the Zassenhaus conjecture for

integral group rings was open for fairly small groups, and so there was an opportu-

nity to raise the lower bound or to find a counterexample . The algebraic study of

association schemes, table algebras, and C-algebras has seen a number of signifi-

cant advances in the last five years, due to the realization of close connections with

finite group theory. As the bases for C-algebras and finite association schemes are

close to finite groups, we felt we could formulate and investigate versions of the

Zassenhaus conjecture in these settings.

The purpose of this dissertation is to shed light upon various properties of

torsion units for both group rings, C-algebras and scheme rings, to verify Zassen-

haus’s conjecture for group rings in some new cases and to move a step toward

establishing the Zassenhaus conjecture for a broader class of rings.

2

The dissertation is organized in the following manner. In Chapter 2, we de-

fine the algebraic structures such as group rings, adjacency algebras and torsion

units of integral C-algebras. We give a brief review of particular known results

for torsion units of integral group rings. In Section 2.1, we define integral group

rings, their normalized torsion units, and we give the statement of the Zassenhaus

conjecture for the normalized torsion units of integral group rings. In Section 2.2,

we define generalized C-algebras and table algebras. In Section 2.3, we define as-

sociation schemes and scheme rings and demonstrate how an association scheme

is a generalization of a group. In Section 2.4, we define semisimple algebras and

state some background results for semisimple algebras that will be used. Then

we define representations and characters for algebras, and give a similar treatment

for groups. In Chapter 3, we state all the preliminary results that will be used in

Chapters 4 and 5 to prove the results. In Section 3.1, we state the results used in

the Luthar-Passi method. In Section 3.2, we demonstrate the Luthar-Passi method

by applying it to the group A4. In Section 3.3, we introduce standard feasible

of a C-algebra that will be needed in Chapters 5. In Chapter 4, we examine the

extent to which rational conjugacy of units in QG is determined by partial aug-

mentations. Also we show that partially central units are not conjugate in QG to

elements of ZG and we construct partially central units of QG for a group of order

3

48. This demonstrates the method we used to verify the Zassenhaus conjecture for

all but three groups of orders up to 159. In Chapter 5, we prove a generalization

of the Berman-Higman Lemma for integral C-algebras and using this we prove a

Lagrange-type theorem for integral C-algebras. We apply these results directly to

several familiar settings, including that of integral scheme rings. We try to move

a step toward generalizing the Zassenhaus conjecture to integral C-algebras by

proving conjugacy of finite subgroups of units of KB in LB implies their conju-

gacy in KB, where K and L are infinite subfields of C with K ⊆ L and B is the

distinguished basis for the C-algebra.

4

Chapter 2

Background

In this chapter we provide background material on group rings, scheme rings, and

C-algebras which we will need in Chapter 4 and Chapter 5 to prove the main

theorems. Basic definitions and concepts of groups, rings, fields, and algebras

that are commonly encountered in undergraduate mathematics courses will be

assumed. These can be found in [11].

2.1 Group rings

Definition 2.1.1. Let R be a commutative ring with identity 1 , 0 and let G be a

finite group. A group ring RG of G over R is a free module over R with basis G,

5

i.e. the set of all formal finite sums

∑αgg, where g ∈ G, αg ∈ R.

The addition in RG is given by:

∑αgg +

∑βgg =

∑(αg + βg)g,

the multiplication in RG by:

(∑

g

αgg)(∑

h

βhh) =∑g, t

(αgβg−1t)t, where t = gh,

and scalar multiplication by:

a(∑g∈G

αgg) =∑g∈G

aαgg, for all a ∈ R.

If we take ring R = Z to be the ring of integers, then the group ring ZG is

called the integral group ring of G. When we take R = Q to be the field of rational

numbers the group ring QG is called the rational group algebra of G. If we take

R = C to be the field of complex numbers, the group ring CG is called the complex

group algebra of G. In general when we take R to be any field then RG is called a

group algebra over R.

If x and g are elements of a group G, we write x ∼ g when x and g are

conjugate in G. Let K(G) be a complete set of representatives for the conjugacy

6

classes of G. When u =∑

g∈Gugg ∈ ZG, ε(u) =

∑g∈G

ug denotes the augmentation

of u, and εx(u) =∑x∼g

ug denotes the partial augmentation of u with respect to

x ∈ G. We note that augmentation is an algebra homomorphism, so the kernel of

this homomorphism is called the augmentation ideal.

A unit in a ring R with identity is any element x ∈ R which has a two-sided

multiplicative inverse y in R. Thus x is a unit of R if and only if for some y ∈ R,

xy = yx = 1. If R is a ring with identity, then U(R) denotes the group of units

of R. A unit of a ring R with finite multiplicative order is called a torsion unit.

The subset of R consisting of torsion units is denoted by U(R)tor. The subgroup of

U(R) consisting of units with augmentation 1 is denoted by V(R). This is the set

of normalized units of R and V(R)tor denotes the set of normalized torsion units of

R. If u =∑

g∈Gugg is a unit in a group ring RG with ug ∈ R, then the support of u is

the set supp(u) = {g ∈ G : ug , 0}. A unit u of a group ring RG is a trivial unit

if u = ugg for some ug ∈ U(R) and a unique element g in the support of u, where

U(R) denotes the group of units of R.

The study of torsion units of integral group rings has centered upon a funda-

mental conjecture made by Hans Zassenhaus in 1966.

Conjecture 2.1.2 (Zassenhaus [32]). Let G be a finite group. Then every element

of V(ZG)tor is rationally conjugate to an element of the group G. That is, if u ∈

7

V(ZG)tor, then there exists b ∈ U(QG) and g ∈ G such that u = b−1gb.

Zassenhaus made several stronger conjectures at the time, all of which have

now been disproven. This one, however, remains open. The Zassenhaus conjec-

ture (abbr. ZC) has been shown to hold in the following general situations:

(i) Abelian groups ([32, Corollary (1.6)]).

(ii) Nilpotent groups (Weiss [35]).

(iii) Groups G with a normal Sylow p-subgroup P for which G/P is abelian

(Hertweck [20]).

(iv) Cyclic-by-abelian groups (Caicedo-Margolis-Del Rio [7]).

(v) All groups of order up to 71. (Hofert-Kimmerle [24]).

(vi) G = A o X, where A and X abelian groups, | X |= m < p for every prime

dividing | A |, and m is prime (Marciniak-Ritter-Sehgal-Weiss [28]).

(vii) Frobenius groups (Hertweck [23]).

8

2.2 Generalized C-algebras and table algebras

In this dissertation we shall consider torsion units of generalized integral C-algebras

and generalized integral table algebras (see [1]). These are finite-dimensional al-

gebras, possibly non-commutative, that have a basis with special properties, one

of which is having integral structure constants. Over the complex numbers these

algebras share many properties in common with the complex group algebra. The

C in C-algebra stands for “character algebra”.

Definition 2.2.1. A (generalized) C-algebra is a triple (A,B, δ), where A is a

finite-dimensional algebra over C with an R-linear and C-conjugate linear in-

volution ∗ : A → A, B = {b0, b1, . . . , bd} is a distinguished basis, δ : A → C

is an algebra homomorphism called the degree map that satisfies the following

properties:

(i) 1A ∈ B (throughout we set b0 to be the multiplicative identity in A),

(ii) (bi)∗ = bi∗ ∈ B, for all bi ∈ B, for a transposition ∗ : {0, 1, . . . , d} →

{0, 1, . . . , d}

(iii) multiplication in A defines real structure constants in the basis B, i.e. for

9

all bi, b j ∈ B, we have

bib j =∑bk∈B

λi jkbk, for some λi jk ∈ R,

(iv) for all bi, b j ∈ B, λi j0 , 0 ⇐⇒ j = i∗,

(v) for all bi ∈ B, λii∗0 = λi∗i0 > 0, and

(vi) δ(bi) = λii∗0 for all bi ∈ B.

If u =∑

i uibi ∈ A with ui ∈ C for all i ∈ {0, 1, . . . , d}, then u∗ =∑

i uibi∗ . The

involution ∗ on A is understood by definition to be an antiautomorphism of A; i.e.

(uv)∗ = v∗u∗, for all u, v ∈ A. Being an algebra homomorphism, the degree map

satisfies δ(uv) = δ(u)δ(v), for all u, v ∈ A. Note also that δ(u∗) = δ(u), for all

u ∈ A.

The real numbers λi jk are the structure constants relative to the basis B. The

C-algebra basis B given in the definition is considered to be standardized because

it satisfies δ(bi) = λii∗0, for all bi ∈ B. A C-algebra (A,B, δ) with standardized

basis B is said to have order n = δ(B+) :=∑

bi∈B δ(bi) and rank r = |B| = d + 1.

The prototype example of a C-algebra is the complex group algebra CG.

We will have use for a few refinements of this definition. A rational (or inte-

gral) C-algebra is a C-algebra whose structure constants in the basis B lie in Q (or

10

respectively, in Z). A table algebra is a C-algebra whose structure constants are

nonnegative. An algebra with a distinguished basis satisfying conditions 1 to 5

of the C-algebra definition are called reality-based algebras (RBAs). RBAs may

or may not have a degree map. If K is an algebraic number field whose ring of

algebraic integers is R, then we will say that a C-algebra (A,B, δ) is R-integral

when all of its structure constants in the basis B lie in the intersection of R with

R. If B is the distinguished basis of a C-algebra (or table algebra), then we shall

say that B is a C-algebra basis (or table algebra basis).

The set of linear elements of a standardized C-algebra basis B is L(B) = {bi ∈

B : bib∗i = λii∗0b0}. Note that b0b∗0 = b20 = b0, so b0 ∈ L(B). Furthermore, if bi ∈ B

then applying the degree map to bib∗i = λii∗0b0 gives δ(bi)2 = δ(bi). Since δ(bi) > 0

we must have λii∗0 = δ(bi) = 1, and so b∗i = (bi)−1. We then can conclude that

L(B) is a finite subgroup of normalized units of ZB.

C-algebras and table algebras have been studied in the commutative situation

in various forms (see Blau’s recent survey [4]). Noncommutative table algebras

were considered in [1] under the name “generalized table algebras”. The alge-

bras considered here are almost the same, except we do not assume the structure

constants are all nonnegative.

11

2.3 Association schemes and scheme rings

We shall only consider finite association schemes, which we now define.

Definition 2.3.1. Let X be a finite set of size n > 0. Let S be a partition of X × X

such that every relation in S is non-empty. For a relation s ∈ S , there corresponds

an adjacency matrix, denoted by σs, which is the n × n (0, 1)-matrix whose (i, j)

entries are 1 if (i, j) ∈ s and 0 otherwise. The pair (X, S ) is an association scheme

if:

(i) S is a partition of X × X consisting of nonempty sets,

(ii) S contains the identity relation 1X := {(x, x) : x ∈ X},

(iii) for all s in S , the adjoint relation s∗ := {(y, x) ∈ X × X : (x, y) ∈ s} also

belongs to S , and

(iv) for all s, t ∈ S , there exist nonnegative integer structure constants astu, for

all u ∈ S , such that σsσt =∑u∈S

astuσu.

A finite association scheme (X, S ), or scheme for short, is said to have order

n = |X| and rank r = |S |. Since S is a partition of X × X, the sum of all adjacency

matrices is the all ones matrix, denoted by J. For notation and background on

association schemes see [38].

12

Examples of association schemes include several familiar algebraic structures.

(i) Finite groups. Let G be a finite group of order n. Let Gτ = {gτ : g ∈ G} ⊆

G × G, where (x, y) ∈ gτ ⇐⇒ xg = y. Then (G,Gτ) is an association

scheme of order n and rank n.

(ii) Schur Rings. Let F be a Schur ring partition of a group G of order n, with

|F | = r. For all U ∈ F , define Uτ ⊂ G × G by (x, y) ∈ Uτ ⇐⇒ xg = y,

for some g ∈ U. Let F τ = {Uτ : U ∈ F }. Then (G,F τ) is an association

scheme of order n and rank r.

(iii) Hecke Algebras. Suppose H is a subgroup of a group G with index n. Let

G/H be the set of left cosets of H in G. Let G//H = {gH : g ∈ G} ⊂

G/H × G/H, where (xH, yH) ∈ gH ⇐⇒ y ∈ xHgH. |G//H| = r is the

number of double cosets of H in G. Then (G/H,G//H) is an association

scheme of order n and rank r.

Our first example above shows that an association scheme is a generalization

of group. But now we give an example of an association scheme that is not a

group.

Example 2.3.2. Let X = {a, b, c, d, e, f } be a set. Let S = {s0, s1, s2, s3} be the

13

relations:

s0 = {(a, a), (b, b), (c, c), (d, d), (e, e), ( f , f )},

s1 = {(a, b), (b, a), (c, d), (d, c), (e, f ), ( f , e)},

s2 = {(a, c), (a, d), (b, c), (b, d), (c, e), (c, f ), (d, e), (d, f ), (e, a), (e, b), ( f , a), ( f , b)},

s3 = {(a, e), (a, f ), (b, e), (b, f ), (c, a), (c, b), (d, a), (d, b), (e, c), (e, d), ( f , c), ( f , d)}.

Then (X, S ) is an association scheme of rank 4 and order 6. The adjacency

matrices are:

σ0 =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

, σ1 =

0 1 0 0 0 0

1 0 0 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

,

σ2 =

0 0 1 1 0 0

0 0 1 1 0 0

0 0 0 0 1 1

0 0 0 0 1 1

1 1 0 0 0 0

1 1 0 0 0 0

, σ3 =

0 0 0 0 1 1

0 0 0 0 1 1

1 1 0 0 0 0

1 1 0 0 0 0

0 0 1 1 0 0

0 0 1 1 0 0

.

and the strucuture constants are given by the following equations:

σ21 = σ0,

σ1σ2 = σ2σ1 = σ2,

σ1σ3 = σ3σ1 = σ3,

σ22 = 2σ3,

σ2σ3 = σ3σ2 = 2σ0 + 2σ1,

σ23 = 2σ2.

14

Note that the collection of adjacency matrices is not a group. The adjacency ma-

trices σ2 and σ3 only have rank 3 and so are not invertible.

Definition 2.3.3. Let (X, S ) be an association scheme and let R be a commutative

ring with identity 1 , 0. The adjacency algebra of the scheme (X, S ) over R is a

ring as well as a free module over R with basis {σs : s ∈ S }, i.e. the set of all

formal finite sums: ∑s∈S

γsσs, where s ∈ S , γs ∈ R.

The addition in RS is given by:

∑s∈S

γsσs +∑s∈S

µsσs =∑s∈S

(γs + µs)σs.

the multiplication in RS is given by:

γµ =∑

s,t

γsµtσsσt =∑s,t,u

γsµtastuσu,

and scalar multiplication by:

a(∑s∈S

γsσs) =∑s∈S

aγsσs, a ∈ R.

So the structure constants of the scheme (X, S ) make the integer span of its

adjacency matrices into a natural Z-algebra ZS := ⊕s∈SZσs. This is known as the

integral adjacency algebra of the scheme (X, S ), which we will simply refer to

15

as the integral scheme ring. Similarly, we can define rational adjacency algebra

(complex adjacency algebra) to be the rational span of adjacency matrices (resp.

complex span of adjacency matrices). Adjacency algebras (integral scheme rings)

are examples of C-algebras (resp. integral C-algebras). Note that the multiplica-

tive identity of ZS is the n × n identity matrix, which is the adjacency matrix

σ1X := σ1.

It is easy to show using the definition of a scheme that the structure constant

ast1 , 0 if and only if t = s∗. We write ns instead of ass∗1 and call ns the va-

lency of s. The linear extension of the valency map defines a degree one algebra

representation CS → C by

u =∑s∈S

usσs → nu =∑s∈S

usns.

So the valency map for scheme rings generalizes the augmentation map for group

rings.

We say that s ∈ S is a thin element of S when ns = 1. The thin radical Oϑ(S )

of S is the subset consisting of the thin elements of S . It follows from the fact that

the valency map is a ring homomorphism that {σt : t ∈ Oϑ(S )} is a group. For the

same reason, the valency of a unit u ∈ U(CS ) has to be a nonzero element of C,

and thus n−1u u is a unit of valency 1. The subgroup of U(ZS ) consisting of units of

valency 1 is denoted by V(ZS ).

16

The next result introduces a natural involution on the adjacency algebras of

finite association schemes.

Proposition 2.3.4. Let (X, S ) be an association scheme. The complex scheme

algebra CS has an involution given as follows: let γ, µ ∈ CS and a ∈ C, for

γ =∑s∈Sγsσs, set γ∗ =

∑s∈Sγsσs∗ , where − denotes the complex conjugate. Then

(i) (γ + µ)∗ = γ∗ + µ∗,

(ii) (γµ)∗ = µ∗γ∗,

(iii) (γ∗)∗ = γ, and

(iv) (aγ)∗ = aγ∗.

Proof. For (i)

(γ + µ)∗ =∑s∈S

(γs + µs)σs∗ =∑s∈S

γsσs∗ +∑s∈S

µsσs∗ = γ∗ + µ∗.

For (ii)

(γµ)∗ = (∑s,tγsµtσsσt)∗ = (

∑s,t,u

γsµtastuσu)∗

=∑s,t,u

γsµtastuσu∗ =∑s,t,u

γsµtat∗s∗u∗σu∗

= (∑µtσt∗)(γsσs∗) = µ∗γ∗.

For (iii)

γ∗ =∑s∈S

γsσs∗ =⇒ (γ∗)∗ =∑s∈S

γsσs = γ.

17

For (iv)

(aγ)∗ = (a∑s∈S

γsσs)∗ = (∑s∈S

aγsσs)∗ =∑s∈S

aγsσs∗ =⇒ (aγ)∗ = aγ∗.

�

It now follows that CS is a generalized integral C-algebra for any finite as-

sociation scheme (X, S ). Its involution is given in Proposition 2.3.4, the set of

adjacency matrices is its C-algebra basis, and the valency map is its degree map.

2.4 Representations and characters of semisimple al-

gebras

2.4.1 Semisimple algebras

Here we collect some basic theory of semisimple algebras.

Theorem 2.4.1 (Krull-Schmidt-Azumaya [10]). Every finite-dimensional algebra

over a field K has a decomposition A = A1 ⊕ · · · ⊕ Ah in which the Ai are inde-

composable two-sided ideals of A. The list of algebras occurring in any such

indecomposable decomposition of A is uniquely determined up to algebra iso-

morphism.

18

Definition 2.4.2. An algebra A is a simple algebra over a field if A , 0 and the

only ideals of A are 0 and A.

Every finite-dimensional simple algebra over a field is isomorphic to a full

matrix algebra over a division algebra over the same field; cf. [10, Theorem 3.28].

Definition 2.4.3. Let K be a field. A finite-dimensional K-algebra A is semisimple

if its indecomposable two-sided ideals are isomorphic to simple K-algebras (i.e.

full matrix rings over K-division algebras).

Definition 2.4.4. Let A be a semisimple algebra over a field K with decomposition

of the following form

A = ⊕mj=1Mn j(K),

where Mn j(K) is full matrix ring of n j × n j matrices over K. Then A is called a

split semisimple algebra.

Definition 2.4.5. Let R be a ring with identity. Then the Jacobson radical J(R) of

a ring R is the intersection of all maximal left ideals of R.

Lemma 2.4.6 (Nakayama’s Lemma). Let R be a ring with identity, and let J(R) be

the Jacobson radical of R. If M is any finitely generated R-module and J(R)M =

M, then M = 0.

19

Proof. Suppose M , 0 and let {m1,m2, . . . ,mn} be a minimal generating set of M

over R. Since M = J(R)M, we have

mn = rlml + r2m2 + ... + rnmn, for some r1, r2, . . . , rn ∈ J(R).

Thus (1 − rn)mn = rlml + ... + rn−1mn−1. It is enough to prove that 1 − rn is a unit.

Suppose 1 − rn is not a unit. Let I be a maximal ideal of R that contains 1 − rn.

Since 1 < I and rn < I we have that rn is not in J(R). Hence 1 − rn is a unit.

�

Theorem 2.4.7. If A is a finite-dimensional algebra over a field K, then J(A) is a

nilpotent ideal; i.e. there exists a positive integer k such that J(A)k = 0.

Proof. Since A is finite-dimensional and J(A)2 ⊆ J(A), there exists a positive

integer m such that J(A)m = J(A)m+i for all positive integers i, in particular

J(A)m = J(A)m+1. Hence J(A)m = J(A)J(A)m, and so by Nakayama’s Lemma,

J(A)m = 0; cf. [11]. �

Proposition 2.4.8 (Theorem 5.18 [10]). A finite-dimensional K-algebra A is semisim-

ple if and only if its Jacobson radical is 0.

In the next theorem we shall reproduce [1, Theorem 3.11] in the setting of

reality-based algebras.

20

Theorem 2.4.9. Let (A,B) be an RBA. Then A is a semisimple algebra.

Proof. Let J be the Jacobson radical of A. Since ∗ is an antiautomorphism of A,

J∗ = J. Assume that J , {0}. Since J is nilpotent, there exists a minimal m ∈ N,

m ≥ 2 such that Jm = {0}. Set I = Jm−1. By the choice of m, I , {0} and I2 = {0}.

Since J∗ = J, I∗ = I. Take a non-zero x ∈ I, where x ∈∑

b∈B xbb, for xb ∈ C. Then

x∗ ∈∑

b∈B xbb∗ ∈ I. Further, xx∗ ∈ I2 = {0}. On the other hand, the coefficient of 1

in the product xx∗ is equal to∑

b∈B λbb∗0xb xb > 0, contradiction. �

Since C-algebras are just RBAs with a degree map, it is immediate from The-

orem 2.4.9 that C-algebras are semisimple algebras. As table algebras and the

adjacency algebras of finite association schemes are C-algebras, these are also

semisimple.

2.4.2 Representation theory of semisimple algebras

Definition 2.4.10. Let K be a field. Let A be a K-algebra. An algebra represen-

tation is a K-algebra homomorphism X : A→ Mn(K).

Definition 2.4.11. Let A be an algebra with basis B = {b0, b1, . . . , bd}. Let {λi jk :

0 ≤ i, j, k ≤ d} be the structure constants relative to the basis B. The representa-

21

tion of A defined by the C-linear extension of

bi 7→ (λi jk)k, j

is called the left regular representation of A.

In the next example we give the left regular representation of a C-algebra.

Example 2.4.12 (Example 4.3 [29]). Let (A,B, δ) be the integral table algebra

whose distinguished basis B = {1 := b0, b1, b2} satisfies

b21 = 2b0 + b1,

b22 = 25b0 + 25b1 + 22b2,

b1b2 = b2b1 = 2b2.

The images of the basis elements in the left regular representation of A are:

b0 =

1 0 0

0 1 0

0 0 1

, b1 =

0 2 0

1 1 0

0 0 2

, b2 =

0 0 25

0 0 25

1 2 22

.The subalgebra of M3(C) generated by these basis matrices is isomorphic to A.

Note that the entries of these basis matrices are precisely the structure constants

relative to the original basis B.

Definition 2.4.13. An algebra representation X of A is irreducible if its image

X(A) is a simple algebra.

22

Definition 2.4.14. Let A be an algebra over a field K. Two representations X1,X2 :

A −→ Mn(K) are equivalent if there is an invertible matrix P ∈ Mn(K) such that

X1(x) = PX2(x)P−1, for all x ∈ A.

Definition 2.4.15. Let X : A −→ Mn(K) be a representation of A. The map

χ : A −→ K given by χ(x) = tr(X(x)), for all x ∈ A is called the character of X.

Also we say that the representation X affords the character χ, and n is the degree

of the representation. Obviously, χ(1) = n, where 1 is the unity element of A.

Definition 2.4.16. Let A be an algebra over a field K. Let X be a representation of

A. If X is an irreducible representation affording the character χ, then χ is called

an irreducible character. The set of all irreducible characters of A is denoted by

Irr(A).

Theorem 2.4.17. The equivalence classes of irreducible representations of a semisim-

ple algebra are distinguished by the distinct irreducible characters.

Proof. Let A be a semisimple algebra over a field K. Let X1 and X2 be equivalent

representations of A of degree n affording the characters χ1 and χ2, respectively.

Then there exists an invertible matrix P ∈ Mn(K) such that X1(x) = PX2(x)P−1,

for all x ∈ A. Since tr(PX2(x)P−1) = tr(X2(x)PP−1) = tr(X2(x)) we have that

χ1(x) = χ2(x), for all x ∈ A. Conversely, let X1 and X2 be two non-equivalent irre-

23

ducible representations affording the same character χ. Therefore X1 and X2 cor-

respond to two different simple A-modules. By [10, Theorem (3.41)], coordinate

functions for both representations must be linearly independent. For any non-zero

x ∈ A, entries of the matrices X1(x) and X2(x) are linearly independent. Therefore,

tr(X1(x)) − tr(X2(x)) , 0, but on the other hand, tr(X1(x)) = χ(x) = tr(X2(x)) for

all x in A. So we reached a contradiction.

�

Let A be a semisimple algebra over a field K and let χ1, χ2, . . . , χm be the ir-

reducible characters of A. If X1 and X2 are representations of A with characters

χ1 and χ2, respectively, then X1 ⊗ X2 affords χ1χ2. Since the product of two char-

acters is a character, C[Irr(A)] is known as a ring of virtual characters. A virtual

character is a character exactly when it lies in N[Irr(A)].

2.4.3 Representation theory of groups

Definition 2.4.18. Let G be a finite group. A representation of G is an algebra

representation of its complex group algebra CG; i.e. an algebra homomorphism

X : CG −→ Mn(C).

If u ∈∑

ugg ∈ CG then X(u) =∑

ugX(g). The characters of representations

24

of CG are referred to as the characters of the group G. Especially important are

the characters of irreducible representations of CG, which carry a great deal of

information about the group G. The set of distinct irreducible characters of the

group G is denoted by Irr(G).

The character table of a group of G is an array, in which the rows are indexed

by the distinct irreducible characters starting with the principal character, the

character that affords the trivial representation, and the columns by the conjugacy

classes of G, starting with the class consisting of the identity element. The value

at the (i, j)-position in the character table is the value of i-th character at j-th

conjugacy class.

Example 2.4.19. Let G = S 3 = {1, (12), (13), (23), (123), (321)} be the symmetric

group on 3 elements. The irreducible representations of G are: the trivial char-

acter, the alternating character, and the 2-dimensional representation of S 3 for

which

(1, 2) 7→

0 1

1 0

, (1, 2, 3) 7→

0 −1

1 −1

This defines an irreducible degree 2 representation because its image on G is

isomorphic to S 3 and the vector space the 6 matrices generate is 4-dimensional.

The character table for symmetric group S 3 is:

25

C1 C2 C3

χ1 1 1 1

χ2 1 −1 1

χ3 2 0 −1

Since C-algebras and adjacency algebras of association schemes are semisim-

ple algebras with distinguished bases, they also have a character theory.

We finish this section with a proof of Kronecker’s theorem; cf. [36, Theorem

4.5.4], which will be needed later on in Chapter 5.

Theorem 2.4.20 (Kronecker). Let α , 0 be an algebraic integer. If α is not a

root of unity, then at least one number Galois conjugate to α has absolute value

strictly grater than 1.

Proof. Let {α1, . . . , αn} be the set of all numbers Galois conjugate to α. Suppose

on the contrary that |αi| ≤ 1, i = 1, . . . , n, and for each k ≥ 0 consider the

polynomial

fk(x) = (x − αk1) · · · (x − αk

n) = xn + ak,n−1xn−1 + · · · + ak,0.

Since α is an algebraic integer, it follows that ak,n−1, , . . . , ak,0 ∈ Z. The con-

dition |αi| ≤ 1 for i = 1, . . . , n imply that |ak,s| ≤nCs, where nCs are the binomial

coefficients. Therefore the coefficients of the polynomials f1, f2, . . . assume only

26

finitely many values, and hence, among these polynomials, there are only finitely

many distinct ones. But then the set of roots of these polynomials is also finite,

and all the numbers α, α2, α3, . . . are in this set. Therefore

αi = α j for some i, j ∈ N and i , j.

Since α , 0, it follows that αi− j = 1. �

27

Chapter 3

Basic Tools

Let ZG be the integral group ring of a finite group G. In the first section of

this chapter we shall present some results that establish a relationship between

torsion units of ZG and partial augmentations. In Section 3.2 we introduce the

Luthar-Passi method, a character-theoretic method that can be used to verify the

ZC for the integral group ring of a specific finite group G. We will demonstrate

the Luthar-Passi method by using it to prove the ZC for the integral group ring of

the alternating group A4. In the last section of the chapter we develop idempotent

formulas and orthogonality relations for C-algebras and association schemes that

will be needed for our discussions in Chapters 4 and 5.

28

3.1 Torsion units of ZG and partial augmentations

First, we will record two foundational results for later reference concerning the

torsion units of ZG. The next lemma of Berman-Higman shows that if e, the

identity of G, lies in the support of a torsion unit of ZG then the unit has to be a

trivial unit.

Lemma 3.1.1 (Corollary (1.3) [33]). Let u =∑

ugg ∈ ZG. If uk = e and ue , 0

then u = uee.

Next we have a Lagrange-type theorem for ZG.

Theorem 3.1.2 (Proposition (1.9) [32]). Let ZG be the integral group ring of a

finite group G. Suppose u ∈ ZG is a normalized torsion unit of order k. Then k is

a divisor of the order of the group G.

The following lemma establishes the relation between the order of an element

of the support of a torsion unit u of ZG and the partial augmentation of u with

respect to that element. This result is deep, it depends on Weiss’ theory of p-

permutation lattices.

Lemma 3.1.3 (Theorem 2.7 [28]). Let u be a normalized torsion unit of ZG with

order k. Let g ∈ G and let p be a prime. If p divides the order of the element g but

not the order of u, then εg(u) = 0.

29

3.2 Luthar-Passi method

The first conjecture of Zassenhaus (ZC) states that every element of V(ZG)tor is

rationally conjugate to an element of group G, where G is a finite group. That is,

if u ∈ V(ZG)tor, then there exists b ∈ U(QG) and g ∈ G such that u = b−1gb. One

computational method for verifying the ZC is the Luthar-Passi method (see [27]).

In this section we will explain and demonstrate how this method works.

Let K(G) be a complete set of representatives for the conjugacy classes of G.

A normalized torsion unit u of QG has trivial partial augmentations if there is a

unique x ∈ K(G) for which εx(u) = 1 and εg(u) = 0 for all other g ∈ K(G).

The idea of the Luthar-Passi method for verifying ZC for a group G is based on

showing that for all k > 1 dividing |G|, the trivial partial augmentations are the

only possible integral solutions of the Luthar-Passi equations for units of order k.

Let U ∈ Mn(C) be a matrix with Uk = 1, k > 1. Then U is a diagonalizable

matrix whose eigenvalues are the k-th roots of unity. Let ζk be a k-th root of unity.

Let µ` be the multiplicity of ζ`k as an eigenvalue of U. Then

Tr(Ur) = µ01 + µ1ζrk + . . . + µk−1ζ

r(k−1)k ,∀1 ≤ r ≤ k.

We get a system of k equations in k unknowns that can be solved by inverting the

30

coefficient matrix, which is of Vandermonde type. The solution is:

µ` =1k

k∑r=1

Tr(Ur)ζ−`rk ` = 0, 1, . . . k − 1.

Let u be a unit in ZG, uk = 1, k ≥ 1. Let χ be any character of G of degree n and

let X be the corresponding representation. The multiplicity µ`(u, χ) of ζ`k as an

eigenvalue of X(u) is given by

µ`(u, χ) =1k

k∑r=1

χ(ur)ζ−`rk ; ` = 0, 1, . . . , k − 1. (3.1)

We will refer to these as the Luthar-Passi equations. Collecting together those r

which have the same gcd with k we have

µ`(u, χ) =1k

∑d|k

∑r′ mod k

d

χ(udr′)ζ−d`r′k .

Since (ud)kd = 1, χ(ud) is a sum of n ( k

d )-th roots ξ1, ξ2, . . . , ξn of unity. Therefore

for (r, kd ) = 1,

χ(udr) = ξ1r + · · · + ξn

r = (χ(ud))σr,

where σr is the automorphism ζdk 7→ ζdr

k of Q(ζdk ). It follows that

µ`(u, χ) =1k

∑d|k

TrQ(ζd)/Q(χ(ud)ζ−d`k ); ` = 0, 1, . . . , k − 1.

By the Luthar-Passi equations from equation (3.1), we have

χ(ur) = µ0(u, χ)1 + µ1(u, χ)ζr + · · · + µk−1(u, χ)ζr(k−1),∀ 1 ≤ r ≤ k,∀ χ ∈ Irr(G).

31

The equations corresponding to a fixed χ can be written in matrix form as

[χ]1×k = [µ]1×k · Fk×k,

where [χ]1×k = [χ(u), χ(u2), χ(u3), . . . , χ(uk)] is the χ-vector,

[µ]1×k = [µ0(u, χ), µ1(u, χ), µ2(u, χ), . . . , µk−1(u, χ)]

is the µ-vector and Fk×k is the Fourier matrix

Fk×k =

1 1 1 . . . 1

ζk ζ2k ζ3

k . . . 1

ζ2k ζ4

k ζ6k . . . 1

......

......

...

ζk−1k ζ2(k−1) ζ3(k−1) . . . 1

The µ-vector is a vector whose entries are nonnegative integers whose sum is n,

the degree of the character. The χ-vector is a matrix whose entries are linear

polynomial expressions in the partial augmentations. We can convert [χ]1×k =

[µ]1×k · Fk×k into a set of linear equations in the partial augmentations and we can

solve these equations. A solution to the system of Luthar-Passi equations is a list

of integer partial augmentations such that for all χ ∈ Irr(G), there is a choice of

µ-vector [µ`(u, χ)]` for which [χ] = [µ] · F.

Example 3.2.1. We will demonstrate the Luthar-Passi method by using it to verify

the ZC for ZA4 for torsion units of order 3. The character table of A4 (see [9, §32])

32

is :g1 g2 g3 g4

χ1 1 1 1 1

χ2 1 1 ζ23 ζ3

χ3 1 1 ζ3 ζ23

χ4 3 −1 0 0

The orders of conjugacy class representatives g1, g2, g3, g4 (in order) are 1, 2, 3,

and 3. Let u =∑

ugg be a normalized torsion unit in ZA4 of order 3, and let

εi = εgi(u), 1 ≤ i ≤ 4,

be the partial augmentations of u. We have for any character χ of A4 of degree n,

χ(u) = ε1χ(g1) + ε2χ(g2) + ε3χ(g3) + ε4χ(g4).

Since the order of u is 3, by Lemma 3.1.1, ε1 = 0, and by Lemma 3.1.3, ε2 = 0.

Therefore, χ(u) = ε3χ(g3) + ε4χ(g4).

Now we generate the Luthar-Passi equations. For ` = 0, 1, 2, we have

µ`(u, χ) = 13

∑d|3

TrQ(ζd3 )/Q(χ(ud)ζ−d`

3 )

= 13 [TrQ(ζ3)/Q(χ(u)ζ−`3 ) + TrQ(ζ3

3 )/Q(χ(u3)ζ−3`3 )]

= 13 [χ(u)ζ−`3 + χ(u)σ2ζ−2`

3 + χ(1)].

Therefore µ`(u, χ) =13

[χ(u)ζ−`3 + χ(u)σ2ζ−2`3 ) + χ(1)], for ` = 0, 1, 2.

33

We know the µ`(u, χ) for 0 ≤ ` ≤ 2 and χ ∈ Irr(G) are the multiplicities of ζ`3 as

an eigenvalue of X(u), where X is the representation affording the character χ. So

all the multiplicities are nonnegative integers and their sum is χ(1), the degree of

the representation X.

Now for χ = χ1,

µ0(u, χ1) =13

[(ε3 + ε4) + (ε3 + ε4) + 1] =13

[2ε3 + 2ε4 + 1],

µ1(u, χ1) =13

[(ε3 + ε4)ζ−13 + (ε3 + ε4)ζ−2

3 + 1] =13

[−ε3 − ε4 + 1],

µ2(u, χ1) =13

[(ε3 + ε4)ζ−23 + (ε3 + ε4)ζ−1

3 + 1] =13

[−ε3 − ε4 + 1].

Since partial augmentations are integers, the only possible µ-vector for χ1 is

[1, 0, 0]. Therefore ε3 + ε4 = 1.

For χ = χ2,

µ0(u, χ2) =13

[(ε3ζ23 + ε4ζ3) + (ε3ζ3 + ε4ζ

23 ) + 1] =

13

[−ε3 − ε4 + 1],

µ1(u, χ2) =13

[(ε3ζ23 + ε4ζ3)ζ−1

3 + (ε3ζ3 + ε4ζ23 )ζ−2

3 + 1] =13

[−ε3 + 2ε4 + 1],

µ2(u, χ2) =13

[(ε3ζ23 + ε4ζ3)ζ−2

3 + (ε3ζ3 + ε4ζ23 )ζ−1

3 + 1] =13

[2ε3 − ε4 + 1].

There are three possible µ-vectors for χ2, [1, 0, 0], [0, 1, 0] and [0, 0, 1]. The µ-

vector [1, 0, 0] produces no compatible solution. The other two µ-vectors [0, 1, 0]

and [0, 0, 1] produce ε3 = 1, ε4 = 0 and ε3 = 0, ε4 = 1 respectively.

34

For χ = χ3,

µ0(u, χ3) =13

[(ε3ζ3 + ε4ζ23 ) + (ε3ζ

23 + ε4ζ3) + 1] =

13

[−ε3 − ε4 + 1].

µ1(u, χ3) =13

[(ε3ζ3 + ε4ζ23 )ζ−1

3 + (ε3ζ23 + ε4ζ3)ζ−2

3 + 1] =13

[2ε3 − ε4 + 1].

µ2(u, χ3) =13

[(ε3ζ3 + ε4ζ23 )ζ−2

3 + (ε3ζ23 + ε4ζ3)ζ−1

3 + 1] =13

[−ε3 + 2ε4 + 1].

Also for χ3, there are three possible µ-vectors, [1, 0, 0], [0, 1, 0] and [0, 0, 1]. A-

gain, the µ-vector [1, 0, 0] produces no compatible solution. The other two µ-

vectors [0, 1, 0] and [0, 0, 1] are consistent with ε3 = 0, ε4 = 1 and ε3 = 1, ε4 = 0

respectively.

For χ = χ4,

µ0(u, χ4) =13

[3] = 1, µ1(u, χ4) =13

[3] = 1, µ2(u, χ4) =13

[3] = 1.

The only possible µ-vector for χ4 is [1, 1, 1], but it does not put any further restric-

tion on the values of ε3 and ε4.

Therefore, there are only two solutions to the Luthar-Passi equations for or-

der 3, and both correspond to trivial units of V(ZA4). The spectral information

corresponding to (ε3, ε4) = (1, 0) is [[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]], and to

(ε3, ε4) = (0, 1) it is [[1, 0, 0], [0, 0, 1], [0, 1, 0], [1, 1, 1]]. The corresponding spec-

tral information proves to be very useful in the new algorithm for constructing

torsion units.

35

3.3 The standard feasible trace of a C-algebra

Definition 3.3.1. Let (A,B, δ) be a C-algebra with standardized basis B. A “fea-

sible trace” is a function ρ : A → C that satisfies the condition ρ(uv) = ρ(vu) for

all u, v ∈ A.

Definition 3.3.2. Let (A,B, δ) be a C-algebra with standardized basis B. The

“standard feasible trace” is the function ρ : A→ C defined by

ρ(∑

i

uibi) = u0δ(B+).

This a feasible trace because ρ(uv) = ρ(vu) for all u, v ∈ A.

Proposition 3.3.3 (Proposition 5.1 [13]). If ρ is a feasible trace for a finite dimen-

sional C-algebra A, then ρ ∈ C[Irr(A)].

By Proposition 3.3.3, there are complex numbers mχ such that ρ =∑χ mχχ,

where the sum runs over χ ∈ Irr(A). When these multiplicities mχ are nonnegative

integers for every χ ∈ Irr(A), ρ is the character of a representation. In this case

we say that ρ is the standard character and any representation affording ρ is called

a standard representation.

When we associate the relations of a finite association scheme (X, S ) of or-

der n with their adjacency matrices, it gives a natural inclusion CS ↪→ Mn(C).

36

This representation of CS is called the standard representation of the association

scheme (X, S ). Its character ρ satisfies ρ(σ0) = n = |X| and ρ(σs) = 0 for all s ∈ S

with s , 1. Clearly the degree of the standard representation is n = |X|.

Now, let (A,B, δ) be a C-algebra, with B = {b0 = 1A, b1, . . . , bd}. By Theorem

2.4.9, A is semisimple, but C is algebraically closed. Therefore (A,B, δ) is a split

semisimple. So

A � ⊕mi=1Ai.

Each Ai is isomorphic to a full matrix algebra over C, of degree ni, say, so

m∑i=1

n2i = r (= d + 1).

Because A is semisimple, any homomorphic image of A is isomorphic to the

complement of its kernel. Therefore, each simple component of A is isomorphic

to the image of an irreducible representation of A. The distinct simple components

of A are in one-to-one correspondence with the equivalence classes of irreducible

representations of A, and hence with the irreducible characters of A. We relabel

the simple components to have

A =⊕χ∈Irr(A)

Aχ.

The identity 1 of A decomposes as 1 =∑χ eχ, with eχ ∈ Aχ, and it is clear that

eχ is a central idempotent of A that is the identity element for the algebra Aχ. The

37

set {eχ : χ ∈ Irr(A)} is a complete set of pairwise orthogonal idempotents of Z(A).

We call these the centrally primitive idempotents of A.

Our next theorem gives the character formula that expresses the centrally prim-

itive idempotents of a C-algebra (A,B, δ) in terms of the distinguished basis B.

Theorem 3.3.4. Let (A,B, δ) be a C-algebra with distinguished basis B = {b0, b1,

. . . , bd}. The centrally primitive idempotent eχ corresponding to χ ∈ Irr(A) can

be expressed as:

eχ =mχ

δ(B+)

d∑i=0

χ(bi∗)λii∗0

bi .

Proof. Let X1, . . . ,Xm be the inequivalent absolutely irreducible representations

of A, and let χ1, . . . , χm be the corresponding characters. The notation will be

chosen so that Xs corresponds to eχs in the sense so that

Xs(eχt) = δstXs(1), 1 ≤ s, t ≤ m.

This implies that Xs has degree χs(1) = χs(eχs) = ns.

Let εχi j for i, j ∈ {1, . . . , χ(b0)} and χ ∈ Irr(A) be a full set of matrix units of

A. Then this set forms a basis for the split semisimple algebra A, and so we can

choose our representations to satisfy

Xχt(εχsi j ) = δstE

χsi j ,

38

where Eχsi j is the elementary matrix with entry δi j in the image of Xχs . These

representations define C-linear coordinate functions aχsi j : A → C for which

Xχs(x) = (aχsi j (x))i j for all x ∈ A, χt ∈ IrrA. Then χt(ε

χsi j ) = δstδi j, so ρ(εχs

i j ) = δi jmχs

and εχsi j ε

χtkl = δstδ jkε

χsil .

Since {εχi j : χ ∈ Irr(A), 1 ≤ i, j ≤ χ(b0)} and {bi : 0 ≤ i ≤ d} are bases for the

algebra A, for some αk ∈ C, we have

εχi j =

d∑k=0

αkbk.

=⇒ εχi j

br∗

λrr∗0=

d∑k=0

αkbkbr∗

λrr∗0=

1λrr∗0

d∑k=0

d∑l=0

αkλkr∗lbl .

=⇒ ρ(εχi jbr∗

λrr∗0) = ρ(

1λrr∗0

d∑k=0

d∑l=0

αkλkr∗lbl) = αrδ(B+).

Since ρ =∑ψ mψψ, where the sum runs over ψ ∈ Irr(A), we have

ρ(εχi jbr∗

λrr∗0) =∑ψ

mψψ(εχi jbr∗

λrr∗0) = mχa

χji(

br∗

λrr∗0).

Therefore

αr =mχ

δ(B+)aχji(

br∗

λrr∗0).

Hence

εχi j =

mχ

δ(B+)

d∑k=0

aχji(bk∗

λkk∗0)bk.

So these primitive idempotents are given by:

εχj j =

mχ

δ(B+)

d∑i=0

aχj j(bi∗)

λii∗0bi .

39

These satisfy∑

j εχj j = eχ and

∑χ

∑j ε

χj j = b0. Hence it follows that the centrally

primitive idempotents of A are:

eχ =mχ

δ(B+)

d∑i=0

χ(bi∗)λii∗0

bi .

�

Corollary 3.3.5. Let (A,B, δ) be a C-algebra with distinguished basis B = {b0, b1,

. . . , bd}. One full set of primitive idempotents for the simple component of A cor-

responding to χ ∈ Irr(A) can be expressed as:

εχj j =

mχ

δ(B+)

d∑i=0

aχj j(bi∗)

λii∗0bi ,

where the maps aχj j are as defined in Theorem 3.3.4.

Any other full set of primitive orthogonal idempotents of A will be conjugate

in A to this set.

The standard feasible trace ρ =∑χ mχχ of a C-algebra need not be a character.

Nevertheless, Blau has recently shown that in fact mχ > 0 for all χ ∈ Irr(A); cf.

[3, Proposition 1]. We shall reproduce this result.

Proposition 3.3.6. Let (A, B, δ) be a C-algebra with the distinguished basis B =

{b0, b1, b2, . . . , bd}. Let ρ =∑χ mχχ be its standard feasible trace, where the sum

runs over χ ∈ Irr(A). Then mχ > 0 for all χ ∈ Irr(A).

40

Proof. Let x ∈∑

xibi ∈ A. Then ρ(xx∗) =∑

xi xiλii∗0δ(B+) implies ρ(eχe∗χ) > 0.

Thus eχe∗χ , 0, so e∗χ = eχ and ρ(eχ) > 0. Since ρ(eχ) =∑

mψψ(eχ) = mχχ(1), this

implies mχ > 0. �

The next corollary is immediate from the last part of the proof of above propo-

sition.

Corollary 3.3.7. Let (A, B, δ) be a C-algebra with the distinguished basis B =

{b0, b1, b2, . . . , bd}. If e is a centrally primitive idempotent then e∗ = e.

We can often use the Theorem 3.3.4 to calculate multiplicities of irreducible

characters in the standard feasible trace of a C-algebra.

Example 3.3.8. Let (A,B, δ) be a C-algebra, where A = M5(C) and let B =

{b0, b1, b2, b3}, where

b1b0 = b1,

b21 = 3b0 + b1 + b2,

b1b2 = b1 + 2b3,

b1b3 = 2b2 + b3.

When we identify B with the image of its regular representation, we have

b1 =

0 3 0 0

1 1 1 0

0 1 0 2

0 0 2 1

, b2 =

0 0 3 0

0 1 0 2

1 0 2 0

0 2 0 1

, b3 =

0 0 0 3

0 0 2 1

0 2 0 1

1 1 1 0

.

41

To calculate the character table for this algebra, it is suffices to simultaneously

diagonalize the three matrices. Since the eigenvalues of matrix b1 are distinct,

using standard linear algebra techniques we can find non-zero eigenvectors gen-

erating each of its eigenspaces. These eigenvectors form the rows of an invertible

matrix Q which diagonalizes b1. The matrix Q in this case is:

Q =

1 1 1 1

1 2/3 −1/3 −2/3

1 −1/3 −1/3 1

1 −2/3 1/3 −2/3

.

Since the algebra is commutative it will also diagonalize b2 and b3. As the

representations of A all have degree 1, the entries on the diagonal of Q−1biQ are

precisely the character values of bi for each bi. Therefore, the character table of A

is:b0 b1 b2 b3

χ1 1 3 3 3

χ2 1 2 −1 −2

χ3 1 −1 −1 1

χ4 1 −2 3 −2

We note the first row is the degree map. Therefore, δ(b0) = 1, δ(b1) = 3 =

δ(b2) = δ(b3), the order δ(B+) = 10, and the rank of A is 4. By Theorem 3.3.4, we

42

know the idempotent corresponding to χ ∈ Irr(A) is:

eχ =mχ

δ(B+)

d∑i=0

χ(bi∗)λii∗0

bi.

Now eχ1 = [mχ1/10](b0 + b1 + b2 + b3), so

[(eχ1)2]0 = [(mχ1)

2/100](b0 + (3b0) + (3b0) + (3b0))0

= [(mχ1)2/100](10b0)0

= (mχ1)2/10.

Since (mχ1)2/10 must be equal to mχ1/10 and mχ1 > 0 so we get mχ1 = 1.

eχ2 = [mχ2/10](b0 + (2/3)b1 − (1/3)b2 − (2/3)b3), so

[(eχ2)2]0 = [(mχ2)

2/100](b0 + (4/9)(3b0) + (1/9)(3b0) + (4/9)(3b0))0

= [(mχ2)2/100](4b0)0

= (mχ2)2/25.

Since (mχ2)2/25 must be equal to mχ2/10 and mχ2 > 0 so we get mχ2 = 5/2.

eχ3 = [mχ3/10](b0 − (1/3)b1 − (1/3)b2 + (1/3)b3), so

[(eχ3)2]0 = [(mχ3)

2/100](b0 + (1/9)(3b0) + (1/9)(3b0) + (1/9)(3b0))0

= [(mχ3)2/100](2b0)0

= (mχ3)2/50.

Since (mχ3)2/50 must be equal to mχ3/10 and mχ3 > 0 so we get mχ3 = 5.

43

eχ4 = [mχ4/10](b0 − (2/3)b1 + (3/3)b2 − (2/3)b3), so

[(eχ4)2]0 = [(mχ4)

2/100](b0 + (4/9)(3b0) + (9/9)(3b0) + (4/9)(3b0))0

= [(mχ4)2/100](20

3 b0)0

= (mχ4)2/15.

Since (mχ4)2/15 must be equal to mχ4/10 and mχ4 > 0 so we get mχ4 = 3/2.

Hence the multiplicities are mχ1 = 1,mχ2 = 5/2,mχ3 = 5, and mχ4 = 3/2.

As these multiplicities are not all integers, this C-algebra has no standard char-

acter.

44

Chapter 4

Normalized Torsion Units of

Integral Group Rings

In the first section of this chapter we prove that for a given group G if two torsion

units of QG are conjugate in QG then they have equal partial augmentations. The

converse holds with some restrictions. In the second and third sections we de-

scribe the computer program we have developed to construct torsion units of CG

with specific partial augmentations from the spectral information accompanying

solutions to the Luthar-Passi equations. In the fourth section we prove a result

about the partially central torsion units QG for finite group G.

45

4.1 Partial augmentations and rational conjugacy

Our first theorem examines the extent to which rational conjugacy of units in QG

is determined by their partial augmentations.

Theorem 4.1.1. Let G be a finite group.

(i) If u, v ∈ QG, and u ∼ v in QG, then εx(u) = εx(v), for all x ∈ G.

(ii) Suppose u and v are torsion units in QG with the same order k > 1. Further

assume that ud ∼ vd for all divisors d of k with 1 < d ≤ k, and that εx(u) =

εx(v) for all x ∈ G. Then u ∼ v in QG.

Proof. (i). Suppose u ∼ v in QG. Then χ(u) = χ(v) for all χ ∈ Irr(G). Let K(G)

be a complete set of representatives for the conjugacy classes of G. Then, for all

χ ∈ Irr(G), we have that

χ(u) = χ(v) =⇒∑

x∈K(G)

εx(u)χ(x) =∑

x∈K(G)

εx(v)χ(x).

Since the character table of a finite group G is an invertible matrix, it follows that

εx(u) = εx(v), for all x ∈ G.

(ii). Let u1 be a torsion unit ofQG with order k. Then u1 produces a specific list

(µ) = (µ`(u1, χ))`,χ of nonnegative integers that are associated with the collection

46

of all the Luthar-Passi equations

µ`(u1, χ) =1k

∑d|k

TrQ(ζdk )/Q(χ(ud

1)ξ−`d)

as ` runs through 0 to k − 1 and χ runs over Irr(G) (see Section 3.2). Our as-

sumptions that χ(ud) = χ(vd) for d > 1 dividing k and that u and v have the same

partial augmentations mean the right hand side of the collection of Luthar-Passi

equations for the unit u matches the right hand side of the Luthar-Passi equations

for the unit v. Therefore, the left hand sides also match, so µ`(u, χ) = µ`(v, χ) for

all ` and χ.

The k nonnegative integers µ`(u1, χ), ` = 0, . . . , k − 1 for a fixed χ, each repre-

sent the multiplicity of ζ`k as an eigenvalue of X(u1), for any irreducible represen-

tation Xwith character χ. Observe that the full spectrum of X(u1) is determined by

these multiplicities. When u1 has finite order, X(u1) is a matrix of finite order, and

is therefore diagonalizable. If χ has degree n, then any matrix in GL(n,C) with

this same spectrum will be conjugate to X(u1) in GL(n,C). In particular, since

µ`(u, χ) = µ`(v, χ) for all `, X(u) is conjugate to X(v) in GL(n,C). Since χ is an

absolutely irreducible character, X(CG) = Mn(C), so in fact we have that X(u) and

X(v) are conjugate in X(CG).

Since this is the case for all irreducible characters χ of G, if we let R be

the regular representation of G, then it must be the case that R(u) and R(v) are

47

conjugate in R(CG). Since R is a faithful representation of G, this implies that u

and v are conjugate in CG.

Since u and v are torsion units of QG, it follows from [32, Lemma 37.5] that u

and v are in fact conjugate in QG (see also Lemma 5.2.11 in the next chapter). �

Some recent observations by Hertweck are particularly useful in improving

performance of computer verifications of the ZC for small groups using the Luthar-

Passi method. We use two of these in particular. The first of these is Lemma 3.1.3.

From this it follows that εx(u) can be nonzero only when o(x) divides o(u). The

second is that if G is a solvable group and u ∈ V(ZG)tor, then G has an element

x for which o(x) = o(u) and εx(u) , 0 (see [22]). Furthermore, Lemma 3.1.1

ensures that the coefficient of the identity in any nontrivial torsion unit of ZG is

always 0. Another property that is useful for dealing with some nontrivial solu-

tions to the Luthar-Passi equations when k has prime power order is a special case

of [8, Theorem 4.1]:

Proposition 4.1.2. Let u ∈ V(ZG)tor, and o(u) = pn. Then the following equations

hold:

(i) 0 ≡ (∑

o(g)=pmug) mod p, for 1 ≤ m < n,

(ii) 1 ≡ (∑

o(g)=pnug) mod p.

48

For further restrictions on partial augmentations see [21].

4.2 Computer implementation of the Luthar-Passi

method

We have written a computer program in GAP [12] that can be used to verify the ZC

for the integral group ring of a small finite group G. The first part is a GAP imple-

mentation of the Luthar-Passi method. It sets up the Luthar-Passi equations using

the character table of the group, and uses a recursive Groebner basis technique to

solve them. The output is a list of partial augmentations with their accompanying

µ-vectors. The second part constructs a unit with this list of partial augmentations

by finding matrices in the image of each irreducible representation whose spec-

trum matches the χ-component of the µ-vector. We continue to use the notation

of Section 3.2.

Let K(G) be a complete set of representatives for the conjugacy classes of G.

A normalized torsion unit u of QG has trivial partial augmentations if there is a

unique x ∈ K(G) for which εx(u) = 1 and εg(u) = 0 for all other g ∈ K(G). The

idea of the Luthar-Passi method for verifying the ZC for G is based on showing

that for all k > 1 dividing |G|, the trivial partial augmentations are the only possible

49

integral solutions of the Luthar-Passi equations for units of order k.

We shall explain our GAP program by examining the units of order k = 6 for

the group G that is identified as SmallGroup(72,40) in the GAP library of small

groups. The group G is a non-split central extension of the form (C3 × C3) : D8.

Our notation for the character table and generators of G was produced using GAP

[12]. The character table for group G=SmallGroup(72,40) is:

1a 2a 2b 2c 3a 4a 6a 6b 3b

χ1 1 1 1 1 1 1 1 1 1

χ2 1 −1 −1 1 1 1 −1 −1 1

χ3 1 −1 1 1 1 −1 −1 1 1

χ4 1 1 −1 1 1 −1 1 −1 1

χ5 2 . . −2 2 . . . 2

χ6 4 −2 . . 1 . 1 . −2

χ7 4 . −2 . −2 . . 1 1

χ8 4 . 2 . −2 . . −1 1

χ9 4 2 . . 1 . −1 . −2

where 2a, 2b, 2c, 3a, 3b, 4a, 6a, 6b are representatives of conjugacy classes (in or-

der) C2,C3,C4,C5,C9,C6,C7,C8 of elements of order 2, 3, 4 and 6.

Step 1: Create the Fourier matrix F = F6×6.

50

Step 2: Create a list of indeterminates xi, one for each conjugacy class of the

group. These will represent the various partial augmentations of the unit u in our

calculations.

Step 3: Determine the list of cases for powers of u of smaller order. This tells

us the conjugacy classes for the group elements ud when d divides k. For our

example, we will have 6 cases:

Case 1 : u3 ∈ C2 and u2 ∈ C5.

Case 2 : u3 ∈ C2 and u2 ∈ C9.

Case 3 : u3 ∈ C3 and u2 ∈ C5.

Case 4 : u3 ∈ C3 and u2 ∈ C9.

Case 5 : u3 ∈ C4 and u2 ∈ C5.

Case 6 : u3 ∈ C4 and u2 ∈ C9.

We proceed to find all solutions to the Luthar-Passi equations of order k in

each case.

Step 4: For each irreducible character χ of G we repeat the following steps.

Step 4a: Create the χ-vector using the character table of the group, the infor-

mation about the particular case, and the list of indeterminates.

For example, the χ-vector for k = 6, χ5 and Case 1 : u3 ∈ C2 and u2 ∈ C5 is:

[2,−2 ∗ x4 + 2 ∗ x5 + 2 ∗ x9, 2, 0, 2,−2 ∗ x4 + 2 ∗ x5 + 2 ∗ x9],

51

where xi is the partial augmentation for the ith conjugacy class in the character

table.

Step 4b: Create a list of possible µ vectors for the multiplicities of various

powers of ζk that could occur for X(u) when X is a representation of G affording

χ.

In our example, since χ5 has degree 2 and all of its character values are ratio-

nal, the possible µ-vectors are:

[2, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [1, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1], [0, 0, 1, 0, 1, 0].

Step 4c, part (i): For each µ-vector in step 4b, find the list of polynomial

equations determined by entries of the matrix equation χ ∗ F−1 − µ = 0.

Step 4c, part (ii): Combine the list of polynomial equations with each Groeb-

ner basis carried forward from previous χ’s, and solve the homogeneous system

formed by these equations by calculating its reduced Groebner basis. When the

solution exists, record the reduced Groebner basis and associated µ-vector. Pass

this information forward to the next character. All chains of valid choices of µ-

vector are carried forward to the next character. Step 4 is concluded once all

characters are exhausted.

We shall demonstrate the full procedure for Case 1 (u3 ∈ C2 and u2 ∈ C5) giv-

ing the polynomials derived from the Luthar-Passi equations for each character

52

and calculating the Groebner basis at each stage. For χ1, since the unit is nor-

malized the µ-vector will be [1, 0, 0, 0, 0, 0]. The six corresponding Luthar-Passi

polynomials are:

1/3 ∗ x2 + 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 + 1/3 ∗ x7 + 1/3 ∗ x8 + 1/3 ∗ x9 − 1/3,

1/6 ∗ x2 + 1/6 ∗ x3 + 1/6 ∗ x4 + 1/6 ∗ x5 + 1/6 ∗ x7 + 1/6 ∗ x8 + 1/6 ∗ x9 − 1/6,

−1/6 ∗ x2 − 1/6 ∗ x3 − 1/6 ∗ x4 − 1/6 ∗ x5 − 1/6 ∗ x7 − 1/6 ∗ x8 − 1/6 ∗ x9 + 1/6,

−1/3 ∗ x2 − 1/3 ∗ x3 − 1/3 ∗ x4 − 1/3 ∗ x5 − 1/3 ∗ x7 − 1/3 ∗ x8 − 1/3 ∗ x9 + 1/3,

−1/6 ∗ x2 − 1/6 ∗ x3 − 1/6 ∗ x4 − 1/6 ∗ x5 − 1/6 ∗ x7 − 1/6 ∗ x8 − 1/6 ∗ x9 + 1/6,

1/6 ∗ x2 + 1/6 ∗ x3 + 1/6 ∗ x4 + 1/6 ∗ x5 + 1/6 ∗ x7 + 1/6 ∗ x8 + 1/6 ∗ x9 − 1/6.The reduced Groebner basis is: x2 + x3 + x4 + x5 + x7 + x8 + x9 − 1.

To save space and time we show the list of polynomials that occur in each case

from now on without repeating scalar multiples.

For χ2 and µ-vector [0, 0, 0, 1, 0, 0] the Luthar-Passi polynomial is uniqe:

−1/3 ∗ x2 − 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 − 1/3 ∗ x7 − 1/3 ∗ x8 + 1/3 ∗ x9 + 1/3.

The reduced Groebner basis from the above polynomial(s) and from the first

reduced Groebner basis is: x4 + x5 + x9, x2 + x3 + x7 + x8 − 1.

If we choose the µ-vector [1, 0, 0, 0, 0, 0] instead of [0, 0, 0, 1, 0, 0] then we get

the trivial ideal. Hence we cannot use this µ-vector for this case.

53

For χ3 and µ-vector [0, 0, 0, 1, 0, 0] the Luthar-Passi polynomial is:

−1/3 ∗ x2 + 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 − 1/3 ∗ x7 + 1/3 ∗ x8 + 1/3 ∗ x9 + 1/3.

The reduced Groebner basis for χ3 corresponding to this sequence of µ-vector

choices is: x4 + x5 + x9, x3 + x8, x2 + x7 − 1.

For χ4 and µ-vector [1, 0, 0, 0, 0, 0] the Luthar-Passi polynomial is:

1/3 ∗ x2 − 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 + 1/3 ∗ x7 − 1/3 ∗ x8 + 1/3 ∗ x9 − 1/3.

The reduced Groebner basis after χ4 corresponding to this sequence of µ-vector

choices is: x4+x5+x9, x3+x8, x2+x7−1. Although the polynomials corresponding

to χ4 for this choice of µ-vector do not change the Groebner basis, we still need to

record and carry forward the µ-vector to the next stage.

For χ5 and µ-vector [1, 0, 0, 1, 0, 0] the Luthar-Passi polynomial is: −2/3∗ x4 +

2/3∗ x5 +2/3∗ x9 and the reduced Groebner basis is: x5 + x9, x4, x3 + x8, x2 + x7−1.

For χ6 and µ-vector [1, 1, 0, 1, 0, 1] the Luthar-Passi polynomial is: −2/3∗ x2 +

1/3 ∗ x5 + 1/3 ∗ x7 − 2/3 ∗ x9 − 1/3 and the reduced Groebner basis is: x7 − x9 − 1,

x5 + x9, x4, x3 + x8, x2 + x9.

For χ7 and µ-vector [0, 1, 1, 0, 1, 1] the Luthar-Passi polynomial is: −2/3∗ x3−

2/3 ∗ x5 + 1/3 ∗ x8 + 1/3 ∗ x9 and the reduced Groebner basis is: x8 + x9, x7 − x9 −

1, x5 + x9, x4, x3 − x9, x2 + x9. .

For χ8 and µ-vector [0, 1, 1, 0, 1, 1] the Luthar-Passi polynomial is: 2/3 ∗ x3 −

54

2/3 ∗ x5 − 1/3 ∗ x8 + 1/3 ∗ x9 and the reduced Groebner basis is: x9, x8, x7 − 1, x5,

x4, x3, x2.

For χ9 and µ-vector [1, 0, 1, 1, 1, 0] the Luthar-Passi polynomial is 2/3 ∗ x2 +

1/3∗ x5−1/3∗ x7−2/3∗ x9 +1/3 and the reduced Groebner basis is: x9, x8, x7−1,

x5, x4, x3, x2.

Therefore corresponding to Case 1, the chain of µ-vector choices

[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0],

[1, 0, 0, 1, 0, 0], [1, 1, 0, 1, 0, 1], [0, 1, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [1, 0, 1, 1, 1, 0].results in the Groebner basis x9, x8, x7 − 1, x5, x4, x3, x2. This is the Groebner basis

for the ideal whose variety consists of the single point (x2, x3, x4, x5, x7, x8, x9) =

(0, 0, 0, 0, 1, 0, 0). In terms of partial augmentations, this means there is only one

nonzero partial augmentation in this solution, which is 1 for the 7-th conjugacy

class.

4.3 Computer construction of torsion units with pre-

scribed partial augmentations

The GAP program for constructing units that we have developed is explained here

using the group G identified as SmallGroup(48,30) in the GAP library. The

55

group G is a non-split central extension of the form C2 : S 4. Our notation for

the character table and generators of G was produced using GAP [12]. There

are 8 columns in GAP’s character table of G corresponding to elements of order

dividing 4:

1a 2a 2b 2c 4a 4b 4c 4d

χ1 1 1 1 1 1 1 1 1

χ2 1 1 1 1 −1 −1 −1 −1

χ3 1 −1 1 −1 −i i −i i

χ4 1 −1 1 −1 i −i i −i

χ5 2 −2 2 −2 0 0 0 0

χ6 2 2 2 2 0 0 0 0

χ7 3 3 −1 −1 −1 −1 1 1

χ8 3 3 −1 −1 1 1 −1 −1

χ9 3 −3 −1 1 −i i i −i

χ10 3 −3 −1 1 i −i −i i

The generators of G in its polycyclic presentation in GAP are: f1 (of order 4), f2

(a central element of order 2), f3 (of order 3), f4 and f5 (both of order 2). The

representatives of conjugacy classes indicated by 2a, 2b, 2c, 4a, 4b, 4c, and 4d (in

order) are f2, f4, f2 f4, f1, f1 f2, f1 f4, and f1 f2 f4.

56

Let X1, . . . ,Xh be a complete list of representatives of the non-equivalent ir-

reducible representations of G. Let χ1, . . . , χh be the irreducible characters of G,

and let ei be the centrally primitive idempotent of CG corresponding to χi for

i = 1, . . . , h.

Since our algorithm records the list of µ-vectors for each irreducible character

χ of G associated with a solution to the Luthar-Passi equations, we know the

spectrum of Xi(u) for any torsion unit of CG that has the partial augmentations

given by that solution. If we can construct a ui ∈ CG for which the spectrum of

Xi(ui) matches the desired spectrum of Xi(u) for each i, then the unit u =∑

i uiei

will have the desired spectrum, and hence the desired partial augmentations.

For each irreducible character χi of G, the steps in the procedure are:

Step 1: Construct a representation Xi of G affording χi. For this purpose we

use the GAP function IrreducibleRepresentationsDixon.

Step 2: Check if there is a gk ∈ K(G) for which the spectrum of ±Xi(gk)

matches that of Xi(u). If so let ui = ±gk.

For example, for the non-trivial partial augmentations [ε4c, ε4a] = [x7, x2] =

[2,−1] of a unit u ∈ ZG of order 4, the µ-vectors corresponding to the irreducible

57

characters of G (in GAP’s ordering) are:

[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 1, 0, 0], [0, 1, 0, 1],

[1, 0, 1, 0], [3, 0, 0, 0], [0, 0, 3, 0], [0, 3, 0, 0], [0, 0, 0, 3].

Let us consider the µ-vector [0, 0, 0, 1] corresponding to irreducible character χ3.

The spectrum corresponding to this µ-vector is [ζ4], so X3(u) = [ζ4]. Since

X3( f1) = ζ4, we can set u3 = f1.

Now consider the µ-vector [0, 3, 0, 0] corresponding to the the irreducible char-

acter χ9. The corresponding spectrum is {ζ4, ζ4, ζ4}, that is,

X9(u) =

ζ4 0 0

0 ζ4 0

0 0 ζ4

.But there is no element in group G whose image under X9 is ζ4I, so we have to

proceed to step 3.

Step 3: Suppose ui has not been found in Step 2.

Step 3a: Find the field of realization F of the representation Xi. The repre-

sentations in GAP are given using their image on the generators of G. So we

only need to calculate the field extension of Q generated by the entries of a few

matrices.

The field of realization depends on the given representation. For our X9, F is

the Gaussian rational field, whereas for X5 it is Q(ζ3).

58

Step 3b: Construct a candidate matrix A of order k from the spectrum of Xi(u).

Use either a diagonal matrix if ζk ∈ F and the factored companion matrix for

the spectrum list otherwise. The diagonal matrices are determined directly from

the i-th component of the µ-vector. The factored companion matrices are the

direct sum of companion matrices corresponding to the irreducible factors of the

characteristic polynomial of the diagonal matrix. We don’t use the companion

matrix of the characteristic polynomial itself because it may not have order k.

The field of realization for our X5 is Q(ζ3), but the spectrum from the µ-vector

[0, 1, 0, 1] is {ζ4, ζ34 }. So we shall use the factored companion matrix

0 −1

1 0

.Since the field of realization for X9 is the Gaussian rational field and the spectrum

from the µ-vector [0, 3, 0, 0] is {ζ4, ζ4, ζ4},

A =

ζ4 0 0

0 ζ4 0

0 0 ζ4

is the required matrix.

Step 3c: Find a subset B of G for which {Xi(b) : b ∈ B} is a Z-basis of Xi(ZG).

First a subset B of G for which Xi(B) is a Q basis of Xi(QG) is found using a

recursive algorithm which adds elements of G to the set B one at a time. If X(g)

lies in the Q-span of X(B), then g is not added to B, and otherwise, it is added.

Then a second recursive algorithm optimizes the choice of elements of G used in

59

B so that every X(g) is a Z-linear combination of the elements {X(b) : b ∈ B}.

Let f1, f2, f3, f4, f5 be the generators (as above) of G. The Z-basis of X9(ZG)

is generated by the images of the subset

B = {Identity, f1, f3, f4, f5, f1 f3, f1 f4, f1 f5, f 23 , f3 f4, f3 f5, f1 f 2

3 ,

f1 f3 f4, f1 f3 f5, f 23 f4, f 2

3 f5, f1 f 23 f4, f1 f 2

3 f5}.

The images of elements of B under X9 are (in order):

1 0 0

0 1 0

0 0 1

,−ζ4 0 −ζ4

0 −ζ4 0

0 0 ζ4

,

0 1 0

1 0 1

1 −1 0

,−1 1 −1

0 0 −1

0 −1 0

,

−1 0 0

0 0 1

0 1 0

,−ζ4 0 0

−ζ4 0 −ζ4

ζ4 −ζ4 0

,ζ4 0 ζ4

0 0 ζ4

0 −ζ4 0

,ζ4 −ζ4 0

0 0 −ζ4

0 ζ4 0

,

1 0 1

1 0 0

−1 1 −1

,

0 0 −1

−1 0 −1

−1 1 0

,

0 0 1

−1 1 0

−1 0 −1

,

0 −ζ4 0

−ζ4 0 0

−ζ4 ζ4 −ζ4

,

60

ζ4 −ζ4 ζ4

ζ4 0 ζ4

−ζ4 ζ4 0

,ζ4 0 0

ζ4 −ζ4 0

−ζ4 0 −ζ4

,−1 0 −1

−1 1 −1

1 0 0

,−1 1 0

−1 0 0

1 −1 1

,

0 0 ζ4

ζ4 −ζ4 ζ4

ζ4 0 0

,

0 0 −ζ4

ζ4 0 0

ζ4 −ζ4 ζ4

.Step 3d: Write A =

∑b∈B abXi(b), ab ∈ F. The coefficients ab are found by sim-

ply writing A as a linear combination of the elements X(b) for b in B. This is ac-

complished in GAP after designating the specific set B as a basis using one of the

options for the Basis command, and then simply asking for Coefficients(B, A)

(see [12]). More meaningful results are sometimes achieved in this step by using

a conjugate of A under a permutation matrix.

In our example, the expression for A our implementation produces for X9 is

A = (−1/2)X9( f1 f3) + (1/2)X9( f1 f4) + (1/2)X9( f1 f5) + (−1/2)X9( f1 f 23 )

+ (−1/2)X9( f1 f3 f5) + (−1/2)X9( f1 f 23 f4).

Since A is a scalar matrix this time, it is reasonable, and desirable for consid-

erations we will encounter in Section 4.4, to expect a result whose coefficients are

constant on conjugacy classes. After manually modifying our choice of the basis

B to favour elements of the class C4a, we obtain

A = −12X9(C4a),

61

where C4a denotes the sum of elements of the conjugacy class C4a.

It is possible that the GAP command Coefficients(B, A) gives no rational

solution. At worst this requires us to switch to an F-basis since there will always

be a solution with coefficients in F.

Step 3e: Set ui =∑

b∈B abb. At this step it uses the coefficients that are calcu-

lated at step 3d and expresses the ui as a linear combination of elements of B. For

example the u9 corresponding to X9 is:

u9 = (−1/2) f1 f3 + (1/2) f1 f4 + (1/2) f1 f5 + (−1/2) f1 f 23

+ (−1/2) f1 f3 f5 + (−1/2) f1 f 23 f4.

Alternatively, u9 can be chosen to be −12C4a.

We now give the results of carrying out steps 1 to 3 in all cases for the group

G. The nontrivial integer solutions to the Luthar-Passi equations for a unit u of

order k = 4 have ε2a = ε2b = ε2c = 0. All of them arise after choosing u2 to lie

in the class 2a. The solutions for the partial augmentations on classes of order 4

come in two patterns:

(ε4a, ε4b, ε4c, ε4d) ∈ {(2, 0,−1, 0), (0, 2, 0,−1), (−1, 0, 2, 0), (0,−1, 0, 2),

(1,−1, 0, 1), (−1, 1, 1, 0), (1, 0, 1,−1), (0, 1, 1,−1)

(0,−1, 1, 1), (−1, 0, 1, 1), (1, 1, 0,−1), (1, 1,−1, 0)}

The lists of multiplicities corresponding to these solutions produce the follow-

62

ing lists of eigenvalues for our candidates for Xχ(u):

χ spec(X(u)) ui

χ1 (1)

χ2 (−1)

χ3 (i), (−i) ← u3 = f1, f1 f2

χ4 (−i), (i)

χ5 (i,−i)

χ6 (1,−1)

χ7 (−1,−1, 1), (−1, 1, 1), (−1,−1,−1), (1, 1, 1) ← u7 = f1,− f1, v,−v

χ8 (−1, 1, 1), (−1,−1, 1), (1, 1, 1), (−1,−1,−1) ← u8 = u7

χ9 (−i,−i,−i), (i, i, i), (−i,−i, i), (−i, i, i) ← u9 = −v, v,− f1, f1

χ10 (i, i, i), (−i,−i,−i), (−i, i, i), (−i,−i, i) ← u9 = u10

The element v = −12C4a is an appropriate scalar multiple of a class sums that

will be mapped onto the desired scalar multiples of the identity matrix by an ir-

reducible representation Xi affording the character χi. Let eχi be the centrally

primitive idempotent of CG corresponding to χi. The idea is that choosing an

appropriate ui ∈ QG for i = 1, . . . , 10 will result in a unit u =10∑i=1

uieχi that has

the desired partial augmentations. It turns out that the choice of u3 will be an ap-

63

propriate choice for u1 through u6, the choice for u7 is appropriate for u8, and the

choice for u9 will work for u10. We have given above 2 choices for u3, 4 choices

for u7, and 4 choices for u9. These options cover all solutions to the Luthar-Passi

equations for order 4 for this group, both trivial and nontrivial.

The torsion units u we construct from the ui will have the form b1e1 + b2e2 +

b3e3, where

e1 = eχ1 + eχ2 + eχ3 + eχ4 + eχ5 + eχ6

e2 = eχ7 + eχ8 , and

e3 = eχ9 + eχ10 .

are nontrivial orthogonal central idempotents of QG that sum to 1, and b1, b2, and

b3 vary among the selections listed above for the ui for i = 3, 7, and 9. The 12

torsion units of order 4 that we get having nontrivial partial augmentations are:

u (ε4a, ε4b, ε4c, ε4d)

f1e1 − ve2 − ve3 (2, 0,−1, 0)

f1e1 + ve2 + ve3 (−1, 0, 2, 0)

f1 f2e1 − ve2 + ve3 (0, 2, 0,−1)

f1 f2e1 + ve2 − ve3 (0,−1, 0, 2)

64

u (ε4a, ε4b, ε4c, ε4d)

f1e1 + f1e2 + ve3 (0, 1, 1,−1)

f1 f2e1 + f1e2 − ve3 (1, 0,−1, 1)

f1e1 − f1e2 − ve3 (1,−1, 0, 1)

f1 f2e1 − f1e2 + ve3 (−1, 1, 1, 0)

f1e1 − ve2 − f1e3 (1, 1, 0,−1)

f1 f2e1 − ve2 + f1e3 (1, 1,−1, 0)

f1e1 + ve2 + f1e3 (0,−1, 1, 1)

f1 f2e1 + ve2 − f1e3 (−1, 0, 1, 1)

A key observation concerning these units is that the ve’s for e ∈ {e2, e3} do not

lie in ZGe. We have checked this with GAP by showing ve does not lie in the

integral span of Ge.

4.4 Partially central torsion units of QG

The form of the units constructed in Section 4.3 occurs frequently among small

group examples. We have discovered a way to show that units of this form do not

produce counterexamples to the ZC. Our reasoning for this fact is based on the

65

following lemma:

Lemma 4.4.1. Let e be a central idempotent ofQG with e , 0, 1. Suppose v ∈ QG

with ve ∈ Z(QG) and ve < ZGe. Then for all t ∈ QG, ve + t(1− e) is not conjugate

in QG to an element of ZG.

Proof. Let t ∈ QG, and let u = ve + t(1 − e). If w is a unit of QG, then uw =

ve+ tw(1−e). Since tw(1−e) always lies in QG(1−e), and QGe∩QG(1−e) = {0},

it cannot make up the difference between ve and any element of ZGe. This means

that uw cannot be an element of ZG, since multiplying it by e does not result in an

element of ZGe. �

The fact that the ZC holds for units of ZG with order 4 seen in Section 4.3

follows from the Lemma 4.4.1 and Theorem 4.1.1.

In the next example we shall show the limitations of our partially central

method by using G:=SmallGroup(72,40) .

Example 4.4.2. Let G be the group SmallGroup(72,40) considered in Section

4.2. The generators of G in its polycyclic presentation in GAP are: f1, f2, f3 (three

of order 2), f4 and f5 (both of order 3). In Case 5 (u3 = C4 and u2 ∈ C5), the

Luthar-Passi method produces a nontrivial solution (ε2, ε4, ε7) = (−1, 1, 1) with

corresponding chain of µ-vectors

66

[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0],

[2, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [0, 0, 1, 2, 1, 0].Let 1 be the identity of G. When we use the algorithm of Section 4.3 to

construct a unit with these partial augmentations, it produces

u1 = u2 = u3 = u4 = u5 = f3 f5,

u6 = (4/3)1 + (−2/3) f4 + (−1) f5 + (−2/3) f 24 ,

u7 = f1 f5 = u8, and

u9 = (ζ3 + 2/3 ∗ ζ23 )1 + (−ζ3 + ζ2

3 ) f1 + (1/3 ∗ ζ3 − 1/3 ∗ ζ23 ) f3

+ (1/3 ∗ ζ3 − 2/3 ∗ ζ23 ) f4 + (1/3 ∗ ζ3 − 1/3 ∗ ζ2

3 ) f5

+ (2/3 ∗ ζ3 − 2/3 ∗ ζ23 ) f3 f5 + (1/3 ∗ ζ3 − 2/3 ∗ ζ2

3 ) f 24 .

Therefore

u = f3 f5(eχ1 + eχ2 + eχ3 + eχ4 + eχ5)

+ u6eχ6 + f1 f5(eχ7 + eχ8) + u9eχ9 .

The last four µ-vectors [2, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [0, 0, 1, 2, 1, 0]

gives us the spectrum (in order) {1, 1, ζ6, ζ56 }, {ζ6, ζ

26 , ζ

46 , ζ

56 }, {ζ6, ζ

26 , ζ

46 , ζ

56 }, {ζ

26 , ζ

36 , ζ

36 , ζ

46 }.

So the last four components are not central.

Therefore, the torsion unit with the given nontrivial partial augmentation does

not satisfy the hypothesis of Lemma 4.4.1, so we can not treat it using the partial

central unit criteria, as it does not have the partially central pattern.

Remark 4.4.3. We have used our computer method to verify ZC for almost all

67

groups of order less than 160, we have shown that any torsion unit ofQG with non-

trivial integral partial augmentations is conjugate to a partially central unit of QG

that is not conjugate to an element of ZG. The exceptions are SmallGroup(72,40)

and its two covering groups of order 144. The previously-known lower bound for

the order of groups known to satisfy ZC was 71 (see [24]). If |G| ≤ 159, and G is

not nilpotent, cyclic by abelian, or an abelian extension of a p-group, or one of the

the three exceptions above, then any torsion unit of QG that produces nontrivial

integral partial augmentations in the Luthar-Passi method is conjugate to a partial-

ly central unit of QG that is not conjugate to an element of ZG. For some groups

of order 160 and 192 not covered by an existing method, it is the Luthar-Passi

method from Section 4.1 that has proved too difficult for our computers to handle.

68

Chapter 5

Torsion units of C-algebras

In this chapter we establish some basic results about torsion units of C-algebras

analogous to what happens for torsion units of group rings. We will consider

torsion units of R-integral C-algebras in Section 5.1, where R is a subring of the

ring of algebraic integers. We obtain results on torsion units for noncommutative

C-algebras having a standard character in Section 5.2, and prove a Lagrange-

type theorem concerning the orders of finite subgroups of torsion units of these in

Section 5.3. These results are similar in spirit to what occurs for integral group

rings, but are also new for integral scheme rings, so we state the results in this

setting as corollaries.

69

5.1 Torsion units of RB

Let (A,B, δ) be an R-integral C-algebra, where R is a subring of C containing the

structure constants generated by the basis B. The R-span of B is a subring of A,

which we will denote by RB. Let Q be the algebraic closure of Q in C, and let

Z be its subring consisting of algebraic integers. For any σ ∈ Gal(Q/Q), note

that the map σ : A −→ A defined by σ(∑

i αibi) =∑

i σ(αi)bi, for all∑

i αibi ∈ A

is an algebra isomorphism as long as σ fixes the structure constants in the basis

B. In particular, since the structure constants of a C-algebra are real, complex

conjugation induces an algebra isomorphism on A that takes∑

i uibi 7→∑

i uibi.

It follows from a recent result of Bangteng Xu that if (A,B, δ) is a commuta-

tive integral C-algebra then all torsion units of ZB are trivial (see [37, Theorem

3.1]). We will focus on the noncommutative case, and investigate the question of

whether the orders of finite subgroups of normalized units of ZB divide the order

of L(B). Related to this is a possible generalization of the ZC on torsion units

to integral scheme rings, which would be that any normalized torsion unit of ZB

should be conjugate in QB to some b ∈ L(B).

The algebraic number α is called totally real if all its conjugates are real. We

begin by showing that part of the proof of Xu’s result holds in the noncommutative

situation.

70

Proposition 5.1.1. Let (A,B, δ) be an integral C-algebra. Let T be the subring of

Z consisting of totally real algebraic integers. Let u be a unit of TB with uu∗ = b0,

where b0 is the identity element. Then u is a trivial unit.

Proof. Let u ∈ A. We have uu∗ = b0 =⇒ 1 = (uu∗)0 =∑

uiuiλii∗0. Since the

C-algebra is integral we have that λii∗0 ≥ 1 for all bi ∈ B, and thus |ui|2 ≤ 1 for all

i ∈ {0, 1, . . . , d}.

If we also have u =∑

i uibi ∈ TB, then for all σ ∈ Gal(Q/Q), σ(ui) = σ(ui)

for all i ∈ {0, 1, . . . , d}. Then 1 = σ(uu∗) = σ(u)σ(u)∗, so the same reasoning as

above tells us |σ(ui)| ≤ 1 for all i and all σ ∈ Gal(Q/Q). By Theorem 2.4.20, the

ui are either 0 or a root of unity.

Since 1 =∑

i |ui|2δ(bi), it must be that exactly one ui is a root of unity, and the

corresponding bi must be a linear element of B. (Since ui is also totally real, we

in fact have shown u = ±bi.) �

Corollary 5.1.2. Let (X, S ) be a finite association scheme. If u is a unit of the

integral scheme ring ZS with uu∗ = σ0 then u is a trivial unit.

A result of Blau which tells us that a primitive idempotent e of a commutative

C-algebra will satisfy e∗ = e; cf. Corollary 3.3.7. If we write a torsion unit u

in the basis {es : s = 0, 1, . . . , d} of primitive idempotents of the commutative

71

C-algebra A, we must have u =∑ζses, where ζs is a root of unity for all s. For an

element u ∈ ZB presented in this way, we have u∗ =∑ζse∗s =

∑ζses = u−1. Then

Proposition 5.1.1 can be applied to show that u is a trivial unit.

Xu’s following theorem makes use of the idea of the above proposition.

Theorem 5.1.3 (Theorem 3.1 [37]). Let (A,B, δ) be a commutative C-algebra,

with B = {b0 = 1A, b1, . . . , bd}. Assume that the structure constants of (A,B, δ) are

rational numbers, and λii∗0 ≥ 1, 0 ≤ i ≤ d. If u =∑d

i=0 uibi is a unit of finite order,

where ui ∈ Z, 0 ≤ i ≤ d, then u = uibi, where ui is a root of unity, and bi is a unit.

Proof. Let u be a unit of finite order. Then u =∑ζses, where es is a primitive

idempotent. Since A is a commutative algebra, we have u∗ =∑ζse∗s =

∑ζses =

u−1, implies uu∗ = b0. Since the structure constants are rational numbers, by

Proposition 5.1.1, u is a trivial unit. �

Corollary 5.1.4. If (A,B, δ) is a commutative integral C-algebra, then every tor-

sion unit of ZB is a trivial unit.

Now suppose (A,B, δ) is a noncommutative R-integral C-algebra. We are in-

terested in characterizing the torsion units of RB up to conjugacy.

Proposition 5.1.5. Let (A,B, δ) be a C-algebra. Suppose u =∑

i uibi ∈ A is a

torsion unit of multiplicative order k. Then u is conjugate in A to a unit v whose

72

b0-coefficient satisfies |v0| ≤ 1, and equality occurs iff u = ζkb0, for some k-th root

of unity ζk ∈ C.

Proof. Let {εχj j : χ ∈ Irr(A), 1 ≤ j ≤ χ(b0)} be the full set of primitive orthogonal

idempotents described above in terms of Corollary 3.3.5. Suppose u is a torsion

unit of A with multiplicative order k. Then its image in the regular representation

of A is diagonalizable, and indeed will be conjugate to a diagonal matrix whose

main diagonal entries are k-th roots of unity. This means that u is conjugate to a

unit v of the form

v =∑χ

∑j

ζχ, jεχj j

for some k-th roots of unity ζχ, j in C.

When we isolate the coefficient of b0 in this presentation of v, we have

v0 =∑χ

∑j

ζχ, jmχ

δ(B+)aχj j(b0) =

1δ(B+)

∑χ

∑j

ζχ, jmχ.

The fact that∑χ

∑j ε

χj j = 1 implies that

∑χ

∑j

mχ =∑χ

mχχ(b0) = δ(B+).

Since the mχ are all positive real, it then follows that |v0| ≤ 1, and equality will

occur if and only if all of the ζχ, j are equal, in which case v = v0b0. (Since this is

central in A, we can conclude that v = u.) �

73

We have the following immediate consequence of Proposition 5.1.5 for central

torsion units (see [32, Cor. 1.7]).

Corollary 5.1.6. Let (A,B, δ) be a C-algebra. Let u =∑

i uibi ∈ A be a central

torsion unit of multiplicative order k. Then the following hold.

(i) |u0| ≤ 1, and equality occurs iff u = ζkb0, for some k-th root of unity ζk ∈ C.

(ii) If the structure constants generated by the basis B are rational and u ∈ ZB,

then either u0 = 0 or u is a trivial unit.

The second part of the above corollary generalizes the Berman-Higman lem-

ma; cf. Lemma 3.1.1.

A C-algebra is symmetric if its involution is trivial on B; i.e. bi = b∗i for all

bi ∈ B. Symmetric C-algebras are automatically commutative, because λi jk =

λ j∗i∗k∗ for all i, j, k ∈ {0, 1, . . . , d}. Since the eigenvalues of a Hermitian matrix

have to be real, the order of a real symmetric or complex Hermitian torsion matrix

can be at most 2. In particular this applies to symmetric association schemes. We

now extend the idea to symmetric C-algebras.

Theorem 5.1.7. Let (A,B, δ) be a symmetric integral C-algebra. Then every tor-

sion unit of ZB is a trivial unit with order at most 2.

74

Proof. Suppose u =∑

i uibi ∈ V(ZB) is a torsion unit. Reasoning as in the proof

of Xu’s result, [37, Theorem 3.1], we have uu∗ = b0. Since (A,B, δ) is a symmetric

integral C-algebra, u∗ = u. Therefore u2 = b0 and u2 = (∑

i u2i λii∗0)b0. Therefore

(u2)0 =∑

i u2i λii∗0 = 1. But for all bi ∈ B, 0 < λii∗0 ∈ Z, therefore, ui can be

nonzero for only one bi, and λii∗0 = 1 for this bi. �

Remark 5.1.8. Since C-algebras are semisimple, the ones of rank 2, 3 or 4 are

automatically commutative. So we know by Xu’s result [37, Theorem 3.1] that

torsion units of ZB will be trivial for (A,B, δ) an integral C-algebra of rank ≤ 4.

We do not yet know if there is a noncommutative C-algebra of rank 5.

5.2 Torsion units for integral C-algebras with a

standard character

Let (A, B, δ) be a C-algebra with the distinguished basis B = {b0, b1, b2, . . . , bd}.

Let λi jk be the structure constants relative to the basis B. In this section we shall

consider C-algebras with standard character; cf. Section 3.3. We know that the C-

algebra (A,B, δ) has a standard character if there is an ∗-algebra homomorphism

Γ : A → Mn(C) of degree n = δ(B+) whose character is equal to the standard

feasible trace ρ. This is equivalent to the condition that all multiplicities mχ for

75

χ ∈ Irr(A) are nonnegative integers. The next lemma proves that a standard

representation of a C-algebra is always a faithful representation.

Lemma 5.2.1. Let (A,B, δ) be a C-algebra that has a standard character. Suppose

Γ is a standard representation of (A,B, δ). Then {Γ(bi) : bi ∈ B} is a linearly

independent set.

Proof. Let B = {b0 = 1, b1, ..., bd} and let {λi jk}i, j,k be the structure constants rel-

ative to the basis B. Suppose∑d

0 µiΓ(bi) = 0 where µi ∈ C. If µ j , 0, then

multiplying both sides of the latter equation by Γ(b j∗) will yield

µ0Γ(b j∗) + µ1Γ(b1b j∗) + . . . + µ jΓ(b jb j∗) + . . . + µdΓ(bdb j∗) = 0.

Applying the trace function to both sides, we get µ jλ j j∗0δ(B+) = 0. As the

values of the degree map of a C-algebra are positive on elements of B, we must

have µ j = 0, which is a contradiction. Therefore, {Γ(bi) : bi ∈ B} is a linear

independent set. �

Our next lemma is an analogue of Berman-Higman’s proposition on torsion

units of group rings; cf. [32, Proposition 1.4].

Lemma 5.2.2. Let (A,B, δ) be an integral C-algebra of order n that has a standard

character. Suppose u is a normalized torsion unit of ZB, and write u =∑

i uibi for

some ui ∈ Z. Then u0 , 0 implies that u = b0.

76

Proof. Let ρ be the standard character of A, so

ρ(bi) =

n if bi = b0,

0 otherwise .

Suppose u has finite order k. Then Γ(u) is a diagonalizable matrix all of whose

eigenvalues are k-th roots of unity. Let spec(Γ(u)) = {ζfik }

ni=1, where ζk is a fixed

primitive k-th root of unity in C. Now ρ(u) =n∑

i=1ζ

fik = u0 · n. Then

|u0|n = |

n∑i=1

ζfik | ≤ n,

and |u0|n = n ⇐⇒ all ζ fik ’s are equal to ζ f1

k = u0.

If σ ∈ Gal(Q/Q), then σ defines an algebra automorphism of A given by

σ(u) =∑

i σ(ui)bi because the structure constants relative to B are integral. So

σ(u) will also be a normalized torsion unit of order k for every σ ∈ Gal(Q/Q),

and σ(u)0 = σ(u0). By the above reasoning, we see that |σ(u)0| ≤ 1 for all

σ ∈ Gal(Q/Q), and so by Theorem 2.4.20, u0 must be either 0 or a root of unity.

When u0 is a root of unity, then all eigenvalues of Γ(u) are equal to this, and we

have Γ(u) = u0 · I = u0Γ(b0).

Since Γ is a faithful representation, we have u = u0 · b0. As δ(u) = 1 we have

u0 = 1 and so u = b0. �

77

Corollary 5.2.3. Let (X, S ) be a finite association scheme. Suppose u =∑s∈S

usσs ∈

V(ZS )tor. Then u1 , 0 implies that u = σ0.

Remark 5.2.4. We note that for a commutative C-algebra Lemma 5.2.2 is a con-

sequence of Corollary 5.1.6.

The conclusion of Lemma 5.2.2 fails if we extend the coefficients beyond rings

of algebraic integers, see the following example for the verification of this claim.

Example 5.2.5. Let (A,B, δ) be the integral table algebra whose distinguished

basis B = {1 := b0, b1, b2} satisfies b21 = b0, b2

2 = 2b0 + 2b1 and b1b2 = b2b1 = b2.

This table algebra is the adjacency algebra of the association scheme of order 4

and rank 3, so it has a standard character. Its adjacency matrices are:

b0 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, b1 =

0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0

, b2 =

0 0 1 1

0 0 1 1

1 1 0 0

1 1 0 0

.Let u = −b0 + b1 + b2. Then u2 = 4b0, so 1

2u is a torsion unit with δ(12u) =

12 (−1+1+2) = 1. Therefore, Z[ 1

2 ]B has nontrivial normalized torsion units whose

support includes b0.

The next proposition shows that if the support of a normalized torsion unit of

ZB includes a linear element that commutes with the unit, then the given unit must

be a trivial unit.

78

Proposition 5.2.6. Let (A,B, δ) be an integral C-algebra that has a standard char-

acter. Let u ∈ V(ZB)tor. If bi ∈ L(B) ∩ supp(u) and bi commutes with u, then

u = bi.

Proof. Let u ∈ V(ZB)tor. Now let bi ∈ L(B) ∩ supp(u) for which bi commutes

with u. Then bi is a unit of ZB and ui , 0. Let u′ = b−1i u. Since bi commutes with

u, u′ has finite order. Since (u′)0 = ui , 0, we must have u′ = b0 by Lemma 5.2.2.

Therefore, u = bi is a trivial unit, as claimed. �

Corollary 5.2.7. Let u ∈ V(ZS )tor. If s ∈ Oϑ(S )∩ supp(u) and σs commutes with

u, then u = σs is a trivial unit of ZOϑ(S ).

The center of the finite association scheme (X, S ) is defined to be Z(S ) = {t ∈

S : σtσs = σsσt, for all s ∈ S }. The scheme (X, S ) is a commutative scheme if

Z(S ) = S . The next corollary is immediate from Corollary 5.2.7.

Corollary 5.2.8. Let (X, S ) be a finite association scheme. Suppose u ∈ V(ZS )tor

is a nontrivial unit. If s ∈ supp(u), then either ns ≥ 2 or s < Z(S ).

There is one further noncommutative situation in which we know central tor-

sion units are trivial.

Theorem 5.2.9. Let (A,B, δ) be an integral C-algebra that has a standard char-

acter. Let p be a prime integer. Suppose that for all bi ∈ B\L(B), λii∗0 is divisible

79

by p. Then every central torsion unit of ZB is a trivial unit.

Proof. Let u be a central torsion unit of ZB with multiplicative order k. Suppose u

is not trivial. By Proposition 5.2.6, the intersection of L(B) with supp(u) must be

empty. Let supp(u) = {b1, . . . , bk} and write u =k∑

i=1uibi. Our assumption implies

that for all bi ∈ supp(u), p divides λii∗0. But then

1 = (uk)0 ∈ spanZ{((u j1b j1)(u j2b j2) . . . (u jkb jk))0 : ji ∈ {1, . . . , k} for 1 ≤ i ≤ k} ⊆ pZ,

a contradiction. �

Corollary 5.2.10. Suppose (X, S ) is a finite association scheme. Suppose there

is a prime integer p that divides ns for every s ∈ S with ns > 1. Then every

normalized central torsion unit of ZS is a trivial unit.

The above corollary applies in particular to the important subclass of p-valenced

association schemes, for which every ns > 1 is a power of the prime p.

The following lemma for C-algebras is an analog of [32, Lemma (37.5)].

Lemma 5.2.11. Let K and F be infinite fields with K ⊆ F. Let (A,B, δ) be a

C-algebra, with B = {b0, b1, . . . , bd} and δ(B+) = n. Suppose that T1 and T2 are

two finite subgroups of units of KB. Then

T1 ∼ T2 in FB =⇒ T1 ∼ T2 in KB.

80

Proof. Let δ(B+) = n, |B| = d, and |T1| = m. Suppose w ∈ U(FB) and that for all

α ∈ T1, αw = βα ∈ T2. We wish to find u ∈ U(KB) with αu = βα. Consider the

equations αu = uβα, and write u =∑

xsbs.

As α runs over T1 we have

αu = uβα ⇐⇒ (∑sαsbs)(

∑t

xtbt) = (∑t

xtbt)(∑sβα,sbs)

⇐⇒∑s

∑tαsxt∑rλstrbr =

∑s

∑t

xtβα,s∑rλtsrbr

⇐⇒∑r

(∑s

∑tαsλstr xt)br =

∑r

(∑s

∑t

xtβα,sλtsr)br

⇐⇒ ([αsλstr]s,t[xt]t) = ([βα,sλtsr]s,t[xt]t), ∀ 0 ≤ r ≤ d.

Thus this reduces to a system of m homogeneous linear equations in d vari-

ables over K of the form Mα,u[xt]t = 0. Let {v1, . . . , v`} be a basis of the solution

space in Kd, with 1 ≤ `. This is also a basis for the solution space in Fd. For

every solution∑i

yivi = [xt]t, the element x =∑t

xtbt will satisfy αx = xβα, for all

α ∈ T1.

Suppose that for every solution [xt] in Kd, the element x =∑t

xtbt is a ze-

ro divisor in KB. Let γ : K` → KB be the map γ(∑i

yivi) = γ([xt]t) =∑t

xtbt.

Consider KB as a subset of Mn(K), and consider the polynomial φ(y1, . . . , y`) =

det(γ(∑i

yivi)) = det(∑t

xtbt). Our assumption implies that this polynomial vanish-

es at infinitely many points in K`, and hence it must be identically zero. However,

the existence of the unit w of FB with T w1 = T2 implies that the polynomial does

81

not vanish on F`, which is a contradiction. It follows that there is at least one point

(y1, . . . , y`) in K` for which∑i

yivi = [xt]t and∑t

xtbt is not a zero divisor in KB.

It then follows that u =∑t

xtbt is a unit of KB for which αu = uβα for all α ∈ T1.

Therefore, T1 and T2 are conjugate in KB. �

This lemma means that in order for two finite subgroups of ZB to be conjugate

in QB, it is enough that they be conjugate in CB. This can be determined using

their spectrum in every irreducible representation, as we have done in our com-

puter implementation of the Luthar-Passi method for integral group rings. What

is missing for this approach in the C-algebra or scheme settings is a reasonable

interpretation of conjugacy classes and partial augmentations.

5.3 Lagrange’s theorem for normalized torsion

units of ZB

The Lagrange theorem for finite groups states that for a finite group G, the order

of any finte subgroup H of G divides the order of group G. This theorem has been

generalized to the case where H is a finite subgroup of V(ZG) for a finite group G

[32, Lemma (37.3)]. In this section we prove this theorem for ZB, where Z is the

ring of algebraic integers and B is a distinguished basis for a Z-integral C-algebra.

82

We begin by extending a result for group algebras over fields of characteristic

0 to C-algebras over fields of characteristic 0.

Theorem 5.3.1. Let (A,B, δ) be a C-algebra that has a standard character. Let

K be a subfield of C containing the structure constants relative to the basis B.

Suppose e is an idempotent of KB, and write e =d∑

i=0eibi with ei ∈ K. Then

e0 = mn ∈ Q, where n = δ(B+) and m is the rank of the image of e in any standard

representation.

Proof. Let Γ be the standard representation and ρ be the standard character of

KB. As e is an idempotent, we know that spec(Γ(e)) = [1(m), 0(n−m)] is a multiset,

where m = rank(Γ(e)). Thus

ρ(e) =

d∑i=0

eiρ(bi) = e0 · n = m.

Therefore, e0 = mn ∈ {0,

1n , . . . ,

n−1n , 1}. Furthermore, e0 = 0 = 0

n ⇐⇒ e = 0 and

e0 = 1 = nn ⇐⇒ e = b0. �

Corollary 5.3.2. Let K be a field of characteristic 0 and (X, S ) be a finite associ-

ation scheme of order n. Let e =∑s∈S

esσs , 0, 1 be a nontrivial idempotent of KS .

Then e1 = mn ∈ Q, 0 < e1 < 1, where n = |X| and m is the rank of e as the matrix

in the standard representation.

83

Corollary 5.3.3. Let K be an algebraic number field with ring of integers R. Sup-

pose (A,B, δ) is an R-integral C-algebra that has a standard character. Then the

only idempotents of RB are 0 and 1.

Proof. Let e ∈ RB be an idempotent. Then e0 ∈ Q+ ∩ R = Z+. By Theorem 5.3.1,

this implies e0 = 0 or 1, and by considering the rank of Γ(e) in these respective

cases we have e = 0 or 1. �

Let G be a finite group, and ZG be its integral group ring. If u =∑

g∈Gugg ∈ ZG

with ug ∈ Z for all g ∈ G, then the augmentation of u is∑

g∈Gug ∈ Z. Viewing G as

a C-algebra basis, we see that L(G) = G, and so the augmentation map coincides

with the degree map. It is a well-known fact about normalized units of group rings

that any finite subgroup of normalized units of ZG is linearly independent in CG;

cf. [32, Lemma (37.1)]. It is natural then to inquire about what happens in the

case of integral C-algebras.

Lemma 5.3.4. Let (A,B, δ) be an integral C-algebra that has a standard charac-

ter. Then any finite group of normalized units of ZB is a set of C-linearly indepen-

dent elements of A.

Proof. Let T = {u1 = b0, u2, . . . , u`} be a finite group of units contained in V(ZB).

Suppose c1ui1 + . . . + ckuim = 0 is an expression of minimal length, where the ui j

84

are elements of T and the coefficients c j ∈ Z are not all 0. Since T is a group,

we can assume without loss of generality that ui1 = b0. Expressing the ui j for

j = 2, . . . ,m, as ui j =∑

s ui j,sbs, we have by Lemma 5.2.2 that ui j,0 = 0 for

j = 2, . . . ,m. It follows that

0 = (c1b0 + c2ui2 + · · · + cmuim)0 = c1,

contradicting the minimal length assumption. Therefore, T is a linearly indepen-

dent set. �

Corollary 5.3.5. Let (X, S ) be a finite association scheme. Then any finite group

of units of valency 1 in ZS is a set of linearly independent elements.

For a finite group G, the order of any finite subgroup H of V(ZG) divides the

order of G (see [32, Lemma (37.3)]). We can give an analogue of this theorem for

integral C-algebras that have a standard character.

Theorem 5.3.6. Let (A,B, δ) be an integral C-algebra that has a standard char-

acter. Suppose that this C-algebra has order n = δ(B+) and size r = |B|. Then the

order of any finite subgroup T of V(ZB) divides n and is at most r.

Proof. Since ZB is a free module with basis B, any linearly independent subset

of ZB has at most r = |B| elements. By Lemma 5.3.4, T is a linearly independent

subset, so |T | ≤ r.

85

Now let e = 1|T |

∑t∈T

t = (T +)/|T |. Since T is a finite group, we have e = e2.

Let Γ be a standard representation of A and let ρ be the standard character. Since

e2 = e , 0, spec(Γ(e)) = [1(m), 0(n−m)], where m is the rank of the matrix Γ(e).

Therefore, ρ(e) = m ∈ Z+. Also ρ(e) = 1|T |ρ(T +) = 1

|T |n(T +)0 = n|T | , since the

argument of Lemma 5.3.4 implies (T +)0 = 1. Therefore m = n|T | , hence |T | divides

n. �

Corollary 5.3.7. Let (X, S ) be a finite association scheme of order n and rank

r. Then the order of any finite subgroup T of V(ZS ) divides n and is at most r.

Symbolically, |T | divides |X| and |T | ≤ |S |.

If H is a subgroup of V(ZG) for a finite group G with |H| = |G|, then ZG = ZH;

cf. [32, Lemma (37.4)]. It is not necessarily true in this case that G be isomorphic

to H as groups. The following lemma proves an analogous result for schemes.

Lemma 5.3.8. Let (X, S ) be a finite association scheme with rank r. If T is a finite

subgroup of V(ZS ) with |T | = r, then ZS = ZT.

Proof. By Lemma 5.3.4, T is linearly independent and thus QS = QT . It follows

that ZS ⊇ ZT and mZS ⊂ ZT for some positive integer m.

Let T = {t1 = σ0, t2, . . . , tr} and let s ∈ S . Then

mσs =

r∑i=1

citi, for some ci ∈ Z.

86

We wish to show that each ci is a multiple of m. For each j ∈ {1, . . . , r}, we have

mσst−1j = c jσ0 +

∑i, j

ci(tit−1j ).

Since by Lemma 5.2.2, (tit−1j )

1= 0 for i , j, the coefficient of σ0 on the right

hand side is c j whereas on the left hand side it is a multiple of m. It follows that

m | c j for j = 1, . . . , r. Therefore, σs ∈ ZT for all s ∈ S , and hence ZS = ZT . �

While thin association schemes give immediate examples where the conclu-

sion of the preceding theorem holds, we are uncertain as to whether ZS can pos-

sess a finite subgroup of normalized units of order |S | when (X, S ) is not thin. The

next example shows that it is certainly possible for the adjacency algebra QS to

be ring isomorphic to a group algebra when (X, S ) is not thin.

Example 5.3.9. Let (X, S ) be the association scheme as27-5 in Hanaki and

Miyamoto’s classification of small association schemes (see [14]). This is a com-

mutative non-symmetric scheme of order 27 and rank 3. We have S = {1X, s, s∗},

where ns = ns∗ = 13, and the structure constants of S are determined by σ2s =

6σs + 7σs∗ , σsσs∗ = 13σ0 + 6σs + 6σs∗ , σ2s∗ = 7σs + 6σs∗ . The character table for

87

the scheme is:

1X s s∗ mχ

χ1 1 13 13 1

χ2 1 −ζ3 + 2ζ23 2ζ3 − ζ

23 13

χ3 1 2ζ3 − ζ23 −ζ3 + 2ζ2

3 13

Analysis of the character table of S shows that QS � QC3, where C3 is a

cyclic group of order 3. Let δ be the irreducible character of CS corresponding

to the valency map, and let ψ, ψ be the other two irreducible characters of CS .

Let {eδ, eψ, eψ} be the centrally primitive idempotents of CS , the character formula

for which can be found in Theorem 3.3.4. An element v of CS with order 3 and

valency 1 is given by

v = eδ + ζ3eψ + ζ23eψ,

and since v is fixed by complex conjugation, v ∈ QS . Using the character formula

for centrally primitive idempotents of CS , we find that

v =19

(−4σ0 − σs + 2σs∗),

and v2 = v∗. So if T = {σ0, v, v2}, then T is a finite subgroup of normalized units

of QS for which QS = QT . In this case ZS ( ZT ⊆ Z[ 19 ]S .

Proposition 5.3.10. Let (A,B, δ) be a C-algebra with standard character and dis-

88

tinguished basis B = {b0, b1} of order n and rank 2. Then

|V(ZB)tor| =

1 if n ≥ 3, and

2 if n = 2.

Proof. Let u be a normalized torsion unit of ZB with multiplicative order k. Our

Lagrange theorem for C-algebras with standard character implies k divides n and

k ≤ 2. So we are done if n is odd. Suppose k = 2. Since B is symmetric and

u ∈ ZB, u = u∗. Therefore, u2 = b0 implies that uu∗ = b0, and so by Proposition

5.1.1, u = bs for some s ∈ B with δ(bs) = 1. Such an element of the C-algebra of

rank 2 with s , 0 only exists when n = 2. �

For symmetric C-algebras of rank 3, we have already seen that normalized

torsion units must be trivial with order at most 2. Non-symmetric association

schemes of rank 3, such as the one seen in the example above, arise naturally

from strongly regular directed graphs.

Proposition 5.3.11. Let (A,B, δ) be a C-algebra with standard character of order

n > 2 and rank 3 with distinguished basis B = {b0, b1, b∗1}. If δ(b1) > 1, then

|V(ZB)tor| = 1.

Proof. Suppose u ∈ V(ZB) is a normalized torsion unit with u , b0. By Lemma

5.2.2, supp(u) = {b1, b∗1}, so u = αb1 + βb∗1 for some α, β ∈ Z. Since δ(u) = 1, we

89

have 1 = αδ(b1) + βδ(b∗1) = (α + β)δ(b1), which is not possible as α, β ∈ Z. �

5.4 Applications to Schur rings and Hecke algebras

Schur rings and integral Hecke algebras (i.e. double coset algebras) are interesting

classical examples of scheme rings. In this section we see how the general results

of Section 5.3 apply in these settings.

5.4.1 Schur rings

Let G be a finite group of order n. Let ZF be a Schur ring defined on the group

G. This means that F is a partition of the set G into nonempty subsets for which

(i) {1G} ∈ F ;

(ii) for all U = {g1, . . . , gk} ∈ F ,U∗ = {g−11 , . . . , g

−1k } ∈ F ; and

(iii) for all U,V,W ∈ F , there exists nonnegative integers λUVW such that

UV =∑W∈F

λUVWW,

where U =∑

g∈U g denotes the sum of the elements of U in the group ring

ZG.

90

The Schur ring ZF is defined to be the Z-span of {U : U ∈ F }, considered as a

subring of ZG. ZF is a free Z-module of rank r = |F |. By extension of scalars

we can consider the Schur ring RF for any commutative ring R. We will refer

to a partition of G with the above properties as a Schur ring partition of G. One

example of a Schur ring partition is the partition F of G into its conjugacy classes,

in which case the complex Schur ring CF is isomorphic to the center of the group

ring CG.

We claim that the Schur ring ZF is isomorphic to an integral scheme ring.

Given the group G and Schur ring partition F , for U ∈ F we set

Uτ = {(x, y) ∈ G ×G : xg = y for some g ∈ U}.

Let F τ = {Uτ : U ∈ F }. Using the properties of the Schur ring partition F , it

is straightforward to show that (G,F τ) is an association scheme of order n = |G|

and rank r = |F |. Furthermore, ZF ' Z[F τ] as rings, where the isomorphism is

produced by the restriction of the regular representation of G to ZF . The restric-

tion of the augmentation map on the group ring to ZF corresponds to the valency

map of Z[F τ] under this isomorphism. The following corollary is the application

of our Lagrange theorem for scheme rings to this special setting.

Corollary 5.4.1. Let F be a Schur ring partition of a finite group G. Then the

order of any finite subgroup of V(ZF ) divides |G| and is at most |F |.

91

5.4.2 Integral Hecke algebras

Let H be a subgroup of a finite group G that has index n. Let G/H be the set of

left cosets of H in G. Let r be the number of distinct double cosets HgH of H in

G for g ∈ G. Corresponding to each double coset HgH for g ∈ G, let

gH := {(xH, yH) : y ∈ xHgH}.

Let G//H := {gH : g ∈ G}. Then (G/H,G//H) is an association scheme of order n

and rank r. This type of association sheme is known as a Schurian scheme, and

its rational adjacency algebra Q[G//H] is ring isomorphic to the ordinary Hecke

algebra eHQGeH, where eH = 1|H|

∑h∈H h. (For details see [15], and note that the

argument given there for this fact does not require that the field be algebraically

closed.) The application of our Lagrange theorem for scheme rings in this special

case gives the next result.

Corollary 5.4.2. Let H be a subgroup of a finite group G that has n left cosets

and r double cosets. Then the order of any finite subgroup of V(Z[G//H]) divides

n and is at most r.

92

Chapter 6

Future Work

Among other results on normalized torsion units of C-algebras we established a

Lagrange-type theorem on normalized torsion units of C-algebras. As this disser-

tation is one of the first treatments of torsion units for integral C-algebras, there is

a great deal of potential for further development along the lines of what has been

done for integral group rings. In the future we shall try to establish a Lagrange-

type theorem for integral C-algebras with no standard character. In our compu-

tational approach, we shall try to raise the bound for small groups satisfying ZC

to 215, because 216 = 2333 is expected to be the next difficult case. Prelimi-

nary results in our computer search show difficulties occur at orders 160, 192 and

200. We would like to know if there is an analog of the Luthar-Passi method for

93

attacking the ZC in the scheme or C-algebra settings.

In the remainder of the chapter we gather preliminary results for these future

investigations.

6.1 Categorical aspects

We define the quasi-direct product of bases of C-algebras analogous to the defini-

tion of quasi-direct product of schemes; cf. [38, Chapter 7].

Lemma 6.1.1. Let (Ai,Bi, δi) be C-algebras with the distinguished bases B1 =

{b0, b1, . . . , bd1} and B2 = {c0, c1, . . . , cd2}. Let B = B1 × B2 = {bi ⊗ cu : bi ∈

B1, cu ∈ B2}. If (bi ⊗ cu)∗ = bi∗ ⊗ cu∗ and other operations are componentwise then

B is a basis for a C-algebra (A,B, δ), where δ(bi ⊗ cu) = δ1(bi) ⊗ δ2(cu).

Proof. Since operations are componentwise, b0 ⊗ c0 is the identity element of A.

For any bi ⊗ cu, b j ⊗ cv ∈ B, let bib j =∑kλi jkbk and cucv =

∑wλuvwcw, where λ′s

are real. Then (bib j ⊗ cucv) =∑kλi jkbk ⊗

∑wλuvwcw implies (bi ⊗ cu)(b j ⊗ cv) =∑

k

∑wλi jkλuvw(bk ⊗ cw). If λ(i,u),( j,v),(k,w) = λi jkλuvw, then for λ(i,u),( j,v),(0,0) , 0, we

have λi j0λuv0 , 0, implies λi j0 , 0, λuv0 , 0. It follows that j = i∗, v = u∗ and

( j, v) = (i∗, u∗). As λ(i,u),(i∗,u∗),(0,0) = λii∗0λuu∗0 and λii∗0 = λi∗i0 > 0, λuu∗0 = λu∗u0 > 0,

so λ(i,u),(i∗,u∗),(0,0) = λ(i∗,u∗),(i,u),(0,0) > 0. Hence (A,B, δ) is a C-algebra. �

94

Example 6.1.2. Let (A,B, δ) be a C-algebra with distinguished basis B = {b0, b1,

. . . , bd}, and let C2 be a cyclic group of order 2, viewed as a distinguished basis of

CC2, a C-algebra with trivial degrees. Then B × C2 is a distinguished basis for a

C-algebra.

The algebraic significance of the quasi-direct product of bases B1 and B2 is

that it is a C-algebra basis of the tensor product algebra of A1 and A2. For the

association scheme and C-algebra settings, this is analogous to the direct product

of groups being a basis for the tensor product of their group algebras: i.e. C[G1 ×

G2] � C[G1] ⊗C[G2].

Definition 6.1.3. Let (A1,B1, δ1) be a C-algebra with distinguished basis B1 =

{b10, b11, . . . , b1d}, and let (A2,B2, δ2) be another C-algebra with distinguished ba-

sis B2 = {b20, b21, . . . , b2n}. A C-algebra homomorphism φ : A1 −→ A2 is an

algebra homomorphism if it satisfies

(i) φ(1) = 1,

(ii) φ(b∗1i) = φ(b1i)∗ for all i = 0, 1, . . . , d,

(iii) for all i = 0, 1, . . . , d, there are nonnegative real numbers µb1i,b2 j such that

φ(b1i) =n∑

j=0µb1i,b2 jb2 j, and

95

(iv) for all b1i, b1 j ∈ B1, supp(φ(b1i))∩ supp(φ(b1 j)) , ∅ implies that there exists

ρb1i,b1 j > 0 such that φ(b1i) = ρb1i,b1 jφ(b1 j).

Definition 6.1.4. Let (X, S ) and (Y,T ) be schemes. A scheme homomorphism from

(X, S ) to (Y,T ) is a pair φ = (φX, φS ) of functions φX : X −→ Y and φS : S −→ T

satisfying:

(i) if (x1, x2) ∈ s then (φX(x1), φX(x2)) ∈ φS (s), and

(ii) for all w, z ∈ X and s ∈ S , if (φX(w), φX(z)) ∈ φS (s) then there exists

(x1, x2) ∈ s such that (φX(x1), φX(x2)) = (φX(w), φX(z)).

Definition 6.1.5. A bijective scheme homomorphism is called an isomorphism.

An scheme isomorphism that preserves structure constants is called an algebraic

isomorphism. A normalized algebraic scheme isomorphism preserves valencies

as well.

6.2 When all units of ZB are trivial

We want to know for which integral C-algebra bases B is it true that V(ZB) =

L(B). This is the “trivial units problem” for integral C-algebras. The next result

is a generalization of [32, Proposition (2.1)] which states that if G is a group and

G∗ = G ×C2 then U(ZG) = ±G =⇒ U(ZG∗) = ±G∗.

96

Proposition 6.2.1. Let (A1,B, δ) be a C-algebra with the distinguished basis B =

{b0, b1, . . . , bd} Let A2 = CC2, where C2 = 〈x〉 is a group of order 2. Then U(ZB) =

±B =⇒ U(Z[B ×C2]) = ±(B ×C2).

Proof. Let α, β, γ, δ ∈ ZB be such that (α + βx)(γ + δx) = 1, then

(αγ + βδ) + (αδ + βγ)x = 1.

Upon comparing coefficients, we have

αγ + βδ = 1, αδ + βγ = 0.

Therefore

(α + β)(γ + δ) = 1, (α − β)(γ − δ) = 1.

Since U(ZS ) = ±S , therefore

α + β = ±u, α − β = ±v, u, v ∈ B.

Thus 2α = ±u ± v. Since the coefficient in left hand side is 2, and the only way

for ±u ± v to have even coefficient is u = ±v. It follows that α = 0 or β = 0. In

any case, α + βx is trivial. �

97

6.3 Normalized automorphisms of QB

Definition 6.3.1. Let (A,B, δ) be a C algebra. An automorphism of QB is a nor-

malized C-algebra homomorphism if it preserves the degree.

The fact that normalized automorphisms of integral group rings preserve class

sums is motivating this line of investigation (see [32, Theorem (36.5)]). The fol-

lowing theorem shows that normalized algebraic isomorphism preserves not only

valencies but also thin radicals under some conditions.

Theorem 6.3.2. Let B1 and B2 be commutative C-algebras of the same order. Let

φ : CB1 −→ CB2 be a normalized C-algebra isomorphism. Then φ(Z[L(B1)]) =

Z[L(B2)] implies that φ(L(B1)) = L(B2).

Proof. Since torsion units of degree 1 are trivial in this case, φ(V(ZB1)tor) =

V(ZB2)tor implies that φ(L(B1)) = L(B2). �

As an application to Corollary 5.2.10, the following theorem is a partial ana-

logue to [32, Theorem (36.3)] for schemes. If S ′ is a closed subset of an associa-

tion scheme then Z(S ′) denotes the center of S ′.

Theorem 6.3.3. Let S and T be p-valanced association schemes for some prime

p with nS and nT finite. Let φ : S −→ T be a normalized algebraic isomorphism,

98

and let its linear extension to ZS also be denoted by φ. If φ : ZS −→ ZT then

Z(Oϑ(S )) ' Z(Oϑ(T )).

Proof. Let φ be the normalized algebraic isomorphism between ZS and ZT . As

φ is normalized, for any σs ∈ Z(Oϑ(S )) with order k, φ(σs) has valency 1, is a

unit of order k, and φ(σs) ∈ Z(ZT ). Since φ(σs) has valency 1, by Corollary

5.2.10, φ(σs) ∈ T , hence lies in Z(Oϑ(S )). Therefore φ(Z(Oϑ(S ))) ⊆ Z(Oϑ(T )).

Conversely, let σt ∈ Z(Oϑ(T )) with order k. Since φ is a normalized isomorphism,

φ−1(σt) ∈ Z(ZS ) and φ−1(σt) has valency 1, so by Corollary 5.2.10, φ−1(σt) ∈ S .

Therefore, φ(Z(Oϑ(S ))) ⊇ Z(Oϑ(T )). Hence φ(Z(Oϑ(S ))) = Z(Oϑ(T )). �

99

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