Computing with Multiplicative Invariants
Euler Symposium at Temple U 04-16-2007
Martin Lorenz
Temple University
Philadelphia
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Apology
Leonhard Euler1707 - 1783
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Overview
Classical invariant theory: a brief introduction
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Overview
Classical invariant theory: a brief introduction
Multiplicative invariants: definition and somecomputational issues
Euler Symposium 04-16-2007 – p. 3/21
Overview
Classical invariant theory: a brief introduction
Multiplicative invariants: definition and somecomputational issues
Regularity: the significance of reflections
Euler Symposium 04-16-2007 – p. 3/21
Overview
Classical invariant theory: a brief introduction
Multiplicative invariants: definition and somecomputational issues
Regularity: the significance of reflections
The Cohen-Macaulay property: exploring the unknown
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Part I: Classical invariant theory
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Algebraic formulation
Given: a linear group
G ⊆ GLn(k)
over some field k (usually C)
Euler Symposium 04-16-2007 – p. 5/21
Algebraic formulation
Given: a linear group
G ⊆ GLn(k)
over some field k (usually C)
G acts on the polynomial algebra
k[x1, . . . , xn]
by "linear substitutions of the variables"
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Algebraic formulation
Given: a linear group
G ⊆ GLn(k)
over some field k (usually C)
G acts on the polynomial algebra
k[x1, . . . , xn]
by "linear substitutions of the variables"
The algebra of (polynomial) invariants is
k[x1, . . . , xn]G = {f ∈ k[x1, . . . , xn] | g(f) = f ∀g ∈ G}
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Geometric view
k[x1, . . . , xn] oo //_______ affine spacekn
��
k[x1, . . . , xn]G oo //________
� ?
OO
quotientkn/G
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Example #1
Linear group: G ={(
1 00 1
)
, g =(
−1 00 −1
)}
action on k[x, y]: g(x) = −x, g(y) = −y
⇓
invariants:
only even degrees:
k[x, y]G = k[x2, y2, xy]
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Example #1
geometric view:
k[x, y] oo //___________ affine plane k2
��
k[x, y]G = k[x2, y2, xy] oo //________
� ?
OO
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Pioneers of invariant theory
Masters of computation:
Aronhold (1819 - 1884)
Clebsch (1833 - 1872)
Gordan (1837 - 1912)
Cayley (1821 - 1895)
Sylvester (1814 - 1897)
Cremona (1830 - 1903)
. . .
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Pioneers of invariant theory
Abstract approach:
David Hilbert1862 - 1943
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Part II: Multiplicative Invariants
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The setting
Given: a group G and a G-lattice L ∼= Zn; so
G→GL(L) ∼= GLn(Z)
an integral representation of G
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The setting
Given: a group G and a G-lattice L ∼= Zn; so
G→GL(L) ∼= GLn(Z)
Choose a base ring k and form the group algebra
k[L] =⊕
m∈L
kxm ∼= k[x±1
1 , . . . , x±1n ] , x
mx
m′
= xm+m′
The G-action on L extends uniquely to a “multiplicative”action by k-algebra automorphisms on k[L]:
g(xm) = xg(m) (g ∈ G,m ∈ L)
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The setting
Given: a group G and a G-lattice L ∼= Zn; so
G→GL(L) ∼= GLn(Z)
Choose a base ring k and form the group algebra
k[L] =⊕
m∈L
kxm ∼= k[x±1
1 , . . . , x±1n ] , x
mx
m′
= xm+m′
The G-action on L extends uniquely to a “multiplicative”action by k-algebra automorphisms on k[L] .
The multiplicative invariant algebra is
k[L]G = {f ∈ k[L] | g(f) = f ∀g ∈ G}
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Example #2
Multiplicative inversion in rank 2: G = 〈g | g2 = 1〉
L = Ze1 ⊕ Ze2
action: g(ei) = −ei
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Example #2
Multiplicative inversion in rank 2: G = 〈g | g2 = 1〉
L = Ze1 ⊕ Ze2
action: g(ei) = −ei
Putting xi = xei we have:
k[L] = k[x±11 , x±1
2 ] with g(xi) = x−1i
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Example #2
Multiplicative inversion in rank 2: G = 〈g | g2 = 1〉
L = Ze1 ⊕ Ze2
action: g(ei) = −ei
Straightforward calculation gives
k[L]G = k[ξ1, ξ2] ⊕ ηk[ξ1, ξ2]
with ξi = xi + x−1i and η = x1x2 + x−1
1 x−12 ; they satisfy
ηξ1ξ2 = η2 + ξ21 + ξ2
2 − 4
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Example #2
Multiplicative inversion in rank 2: G = 〈g | g2 = 1〉
L = Ze1 ⊕ Ze2
action: g(ei) = −ei
Hence: k[L]G ∼= k[x, y, z]/(x2 + y2 + z2 − xyz − 4)
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Some computational issues
Back to general multiplicative actions:
G a finite groupL a G-latticek a commutative base ringk[L] the group algebra
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Some computational issues
In general, k[L] has no grading ( with only k in degree 0)that is preserved by the action of G
computational theory not yet highly developed∃ some GAP & MAGMA-programs (L., Marc Renault)
But . . .
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Some computational issues
Jordan (1880): GLn(Z) has only finitely many finite subgroupsup to conjugacy
there are only finitely many multiplicative invariantalgebras k[L]G (up to ∼=) with rank L bounded
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Some computational issues
Jordan (1880): GLn(Z) has only finitely many finite subgroupsup to conjugacy
Minkowski (1887): The least common multiple of their ordersis given by
Mn =∏
p
p⌊ n
p−1⌋+j
n
p(p−1)
k
+j
n
p2(p−1)
k
+...
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Some computational issues
n # fin. G ≤ GLn(Z) # max’l G Mn
(up to conj.) (up to conj.)
1 2 1 2
2 13 2 24
3 73 4 48
4 710 9 5760
5 6079 17 11520
6 85311 39 2903040
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Some computational issues
Accessing these groups via computer:
All finite subgroups G ≤ GLn(Z) with n ≤ 4 areavailable through GAP
GAP and MAGMA both have data bases of allmaximal finite subgroups of GLn(Z) for n ≤ 31
The specialized computer algebra system CARATprovides all finite G ≤ GLn(Z) (and more) up to n = 6
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Part III: Regularity
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Regularity and reflections
Notations: G a finite groupL ∼= Zn a faithful G-latticek = k a field with char k ∤ |G|
Will explain the following result . . .
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Regularity and reflections
Theorem 1 TFAE (1) k[L]G is regular(2)G acts as a reflection group on L
and H1(G/D, LD) = 0
(3) k[L]G ∼= k[Zr+ ⊕ Zs]
(4) ∃ root system Φ s.t. L/LG ∼= Λ(Φ)
and G = W(Φ)
Here, D is the subgroup of G that is generated by the“diagonalizable” reflections, conjugate in GL(L) to
d =
(
−11
...1
)
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Regularity and reflections
A search of the crystallographic groups libraryin GAP yields
n# finite G ≤ GLn(Z)
(up to conjugacy)# reflection groups G
(up to conjugacy)# cases withk[L]G regular
2 13 9 7
3 73 29 18
4 710 102 51
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Example #3: the root latticeAn−1
Notation: Un =⊕n
1 Zei∼= Zn
An−1 = {∑
i ziei ∈ Un |∑
i zi = 0} ∼= Zn−1
Sn-action: σ(ei) = eσ(i) (σ ∈ Sn)
Note: Sn acts as a reflection group;transpositions are reflections
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Example #3: the root latticeAn−1
Notation: Un =⊕n
1 Zei∼= Zn
An−1 = {∑
i ziei ∈ Un |∑
i zi = 0} ∼= Zn−1
Sn-action: σ(ei) = eσ(i) (σ ∈ Sn)
C[An−1]Sn is not regular:
(n > 2; picture for n = 3)
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Part IV: The Cohen-Macaulay Property
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CM Rings
Hypotheses: R a comm. noetherian ringa an ideal of R
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CM Rings
Hypotheses: R a comm. noetherian ringa an ideal of R
Always:
height a ≥ depth a = inf{i | H ia(R) 6= 0}
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CM Rings
Hypotheses: R a comm. noetherian ringa an ideal of R
Always:
height a ≥ depth a = inf{i | H ia(R) 6= 0}
(Zariski) topologydimension theory
(homological) algebra
Def: R is Cohen-Macaulay iff equalityholds for all (maximal) ideals a
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Some Examples of CM Rings
Standard example: R an affine domain/PID k, finite /some polynomial subalgebra P = k[x1, . . . , xn]. Then:
R CM ⇔ R is free over P
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Some Examples of CM Rings
Standard example: R an affine domain/PID k, finite /some polynomial subalgebra P = k[x1, . . . , xn]. Then:
R CM ⇔ R is free over P
Hierarchy: catenary
regular +3 completeT +3 Gorenstein +3 CM
KS
dim 0
5=rrrrrrrrrrr
rrrrrrrrrrr dim 1reduced
KS
dim 2normal
`h HHHHHHHH
HHHHHHHH
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Multiplicative Invariants: CM-property
Notations: G is a finite group 6= 1
L a G-lattice, WLOG faithful
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Multiplicative Invariants: CM-property
Notations: G is a finite group 6= 1
L a G-lattice, WLOG faithful
If |G|−1 ∈ k ("non-modular case") then k[L]G is certainly CM;otherwise usually not
Will concentrate on k = Z
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Multiplicative Invariants: CM-property
Notations: G is a finite group 6= 1
L a G-lattice, WLOG faithful
Theorem 2(L, TAMS ’06)
If Z[L]G is CM then all Gm/R2(Gm) for m ∈ L
are perfect groups, but not all Gm are.
subgroup gen. bybi-reflections on L:rank(gL − IdL) ≤ 2
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Multiplicative Invariants: CM-property
Notations: G is a finite group 6= 1
L a G-lattice, WLOG faithful
Theorem 2(L, TAMS ’06)
If Z[L]G is CM then all Gm/R2(Gm) for m ∈ L
are perfect groups, but not all Gm are.
Corollary (“3-copies conjecture”) Z[L⊕r]G is never CMfor r ≥ 3.
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Test case: the groupSn
Problem What are the Sn-lattices L
such that Z[L]Sn is CM ?
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Test case: the groupSn
Problem What are the Sn-lattices L
such that Z[L]Sn is CM ?
Theorem 2 and classification results for certain finite lineargroups (Huffman and Wales, 70s) allow to reduce this problem to thefollowing question about polynomial invariants . . .
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Test case: the groupSn
Problem’(open for p ≤ n/2)
Are the "vector invariants"Fp[x1, . . . , xn, y1, . . . , yn]Sn CM?
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Test case: the groupSn
Problem’(open for p ≤ n/2)
Are the "vector invariants"Fp[x1, . . . , xn, y1, . . . , yn]Sn CM?
1st open case n = 4, p = 2 is ok! (Thanks to MAGMA)
> n:=4; p:=2;
> G:=PermutationGroup< 2*n | (i,i+1)(n+i,n+i+1) : i in [1..n-1] >;
> R:=InvariantRing(G,GF(p));
> IsCohenMacaulay(R);
true
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Test case: the groupSn
Problem’(open for p ≤ n/2)
Are the "vector invariants"Fp[x1, . . . , xn, y1, . . . , yn]Sn CM?
1st open case n = 4, p = 2 is ok! (Thanks to MAGMA)
It follows that Problem’ is ok for n ≤ 5 (easy argument)
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Test case: the groupSn
Problem’(open for p ≤ n/2)
Are the "vector invariants"Fp[x1, . . . , xn, y1, . . . , yn]Sn CM?
1st open case n = 4, p = 2 is ok! (Thanks to MAGMA)
It follows that Problem’ is ok for n ≤ 5 (easy argument)
Next open cases n = 6, p = 2 and p = 3: out of memory!
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Summary
Let L be a G-lattice, where G is a finite group.
G is generated byreflections on L
Bourbaki, Farkas
L.*2 Z[L]G is asemigroup algebra
?
jr
Hochster
��
G is generated bybireflections on L
Z[L]G isCohen-Macaulay
?
ks
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