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An Introduction to Invariant Theory
Harm Derksen, University of Michigan
Optimization, Complexity and Invariant TheoryInstitute for Advanced Study, June 4, 2018
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Plan of the Talk
I applications of invariants
I a classical, motivating example : binary forms
I polynomial rings ideals
I group representations and invariant rings
I Hilbert’s Finiteness Theorem
I the null cone and the Hilbert-Mumford criterion
I degree bounds for invariants
I polarization of invariants and Weyl’s Theorem
I Invariant Theory for other fields
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Applications of Invariants
Definition
an invariant is a quantity or expression that stays the same undercertain operations
the total energy in a physical system is an invariant as the systemevolves over time
loop invariants can be used to prove the correctness of an algorithmalthough the number of iterations in a loop may vary, the loopinvariant tell us to say something about the variables after theiterations
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Applications of Invariants
Definition
an invariant is a quantity or expression that stays the same undercertain operations
the total energy in a physical system is an invariant as the systemevolves over time
loop invariants can be used to prove the correctness of an algorithmalthough the number of iterations in a loop may vary, the loopinvariant tell us to say something about the variables after theiterations
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Applications of Invariants
Definition
an invariant is a quantity or expression that stays the same undercertain operations
the total energy in a physical system is an invariant as the systemevolves over time
loop invariants can be used to prove the correctness of an algorithmalthough the number of iterations in a loop may vary, the loopinvariant tell us to say something about the variables after theiterations
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Applications of Invariants
Knot invariants (such as the Jones polynomial) can be used todistinguish knots
knot invariants remain unchanged under Reidemeister moves
(co-)homology groups are invariants of topological manifolds
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Applications of Invariants
Knot invariants (such as the Jones polynomial) can be used todistinguish knots
knot invariants remain unchanged under Reidemeister moves
(co-)homology groups are invariants of topological manifolds
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Invariant Theory
in invariant theory we restrict ourselves to
I invariants that are polynomial functions on a vector space
I invariants that remain unchanged under group symmetriessuch as rotations, permutations etc.
we start with a motivating example from 19th century invarianttheory
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Invariant Theory
in invariant theory we restrict ourselves to
I invariants that are polynomial functions on a vector space
I invariants that remain unchanged under group symmetriessuch as rotations, permutations etc.
we start with a motivating example from 19th century invarianttheory
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Invariant Theory
in invariant theory we restrict ourselves to
I invariants that are polynomial functions on a vector space
I invariants that remain unchanged under group symmetriessuch as rotations, permutations etc.
we start with a motivating example from 19th century invarianttheory
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
a binary form of degree 2 is a polynomial
p(z ,w) = p1z2 + p2zw + p3w
2
with p1, p2, p3 ∈ C
SL2 =
{(a bc d
): ad − bc = 1
}is the group of 2× 2 matrices with determinant onea matrix A ∈ SL2 gives a linear change of coordinates in C2
the group SL2 acts on (the coefficients of) binary forms:we make the substitution (z ,w) 7→ (az + cw , bz + dw) and getanother polynomial
p′(z ,w) = p(az + cw , bz + dw) = p′1z2 + p′2zw + p′3w
2
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
a binary form of degree 2 is a polynomial
p(z ,w) = p1z2 + p2zw + p3w
2
with p1, p2, p3 ∈ C
SL2 =
{(a bc d
): ad − bc = 1
}is the group of 2× 2 matrices with determinant onea matrix A ∈ SL2 gives a linear change of coordinates in C2
the group SL2 acts on (the coefficients of) binary forms:we make the substitution (z ,w) 7→ (az + cw , bz + dw) and getanother polynomial
p′(z ,w) = p(az + cw , bz + dw) = p′1z2 + p′2zw + p′3w
2
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
a binary form of degree 2 is a polynomial
p(z ,w) = p1z2 + p2zw + p3w
2
with p1, p2, p3 ∈ C
SL2 =
{(a bc d
): ad − bc = 1
}is the group of 2× 2 matrices with determinant onea matrix A ∈ SL2 gives a linear change of coordinates in C2
the group SL2 acts on (the coefficients of) binary forms:we make the substitution (z ,w) 7→ (az + cw , bz + dw) and getanother polynomial
p′(z ,w) = p(az + cw , bz + dw) = p′1z2 + p′2zw + p′3w
2
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
where
A =
(a bc d
)∈ SL2,p′1
p′2p′3
= MA
p1p2p3
and
MA =
a2 ab b2
ac ad + bc bdc2 cd d2
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
the polynomial f (x1, x2, x3) = x22 − 4x1x3 ∈ C[x1, x2, x3] (thediscriminant) can be viewed as a function from C3 to C and aneasy calculation shows that
f
p1p2p3
= p22 − 4p1p3 = (p′2)2 − 4p′1p′3 = f
p′1p′2p′3
we say that f (x1, x2, x3) is an invariant under the action of SL2
f (x1, x2, x3) is a fundamental invariant that generates all invariants:if h(x1, x2, x3) is another polynomial invariant, then there exists apolynomial q(y) such that h(x1, x2, x3) = q(f (x1, x2, x3))
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
the polynomial f (x1, x2, x3) = x22 − 4x1x3 ∈ C[x1, x2, x3] (thediscriminant) can be viewed as a function from C3 to C and aneasy calculation shows that
f
p1p2p3
= p22 − 4p1p3 = (p′2)2 − 4p′1p′3 = f
p′1p′2p′3
we say that f (x1, x2, x3) is an invariant under the action of SL2
f (x1, x2, x3) is a fundamental invariant that generates all invariants:if h(x1, x2, x3) is another polynomial invariant, then there exists apolynomial q(y) such that h(x1, x2, x3) = q(f (x1, x2, x3))
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
the polynomial f (x1, x2, x3) = x22 − 4x1x3 ∈ C[x1, x2, x3] (thediscriminant) can be viewed as a function from C3 to C and aneasy calculation shows that
f
p1p2p3
= p22 − 4p1p3 = (p′2)2 − 4p′1p′3 = f
p′1p′2p′3
we say that f (x1, x2, x3) is an invariant under the action of SL2
f (x1, x2, x3) is a fundamental invariant that generates all invariants:if h(x1, x2, x3) is another polynomial invariant, then there exists apolynomial q(y) such that h(x1, x2, x3) = q(f (x1, x2, x3))
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
we may identify binary forms of degree n with vectors in Cn+1:
p1zn + p2z
n−1w + · · ·+ pn+1wn ↔
p1p2...
pn+1
the vector space of binary forms of degree n is an(n + 1)-dimensional representation of SL2
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
we may identify binary forms of degree n with vectors in Cn+1:
p1zn + p2z
n−1w + · · ·+ pn+1wn ↔
p1p2...
pn+1
the vector space of binary forms of degree n is an(n + 1)-dimensional representation of SL2
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
polynomial invariants for binary forms of arbitrary degree wereextensively studied in the 19th century by mathematicians likeBoole, Sylvester, Cayley, Aronhold, Hermite, Eisenstein, Clebsch,Gordan, Lie, Klein, Capelli etc.
Theorem (Gordan 1868)
for binary forms of degree d there exists a finite system offundamental invariants that generate all invariants (i.e., everyinvariant is a polynomial expression in the fundamental invariants)
one of the main objectives was to find an explicit system offundamental invariants for binary forms up to degree d
(currently known for d ≤ 10)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
polynomial invariants for binary forms of arbitrary degree wereextensively studied in the 19th century by mathematicians likeBoole, Sylvester, Cayley, Aronhold, Hermite, Eisenstein, Clebsch,Gordan, Lie, Klein, Capelli etc.
Theorem (Gordan 1868)
for binary forms of degree d there exists a finite system offundamental invariants that generate all invariants (i.e., everyinvariant is a polynomial expression in the fundamental invariants)
one of the main objectives was to find an explicit system offundamental invariants for binary forms up to degree d
(currently known for d ≤ 10)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Classical Invariant Theory: Binary Forms
polynomial invariants for binary forms of arbitrary degree wereextensively studied in the 19th century by mathematicians likeBoole, Sylvester, Cayley, Aronhold, Hermite, Eisenstein, Clebsch,Gordan, Lie, Klein, Capelli etc.
Theorem (Gordan 1868)
for binary forms of degree d there exists a finite system offundamental invariants that generate all invariants (i.e., everyinvariant is a polynomial expression in the fundamental invariants)
one of the main objectives was to find an explicit system offundamental invariants for binary forms up to degree d
(currently known for d ≤ 10)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Polynomial Ring
x1, x2, . . . , xn coordinate functions on V = Cn
a polynomial f (x1, . . . , xn) can be viewed as function from V to CC[x] = C[x1, . . . , xn] graded ring of polynomial functions
Definition (Ideal)
a subset I ⊆ C[x] is an ideal if
1. 0 ∈ I ;
2. f (x), g(x) ∈ I ⇒ f (x) + g(x) ∈ I ;
3. f (x) ∈ C[x], g(x) ∈ I ⇒ f (x)g(x) ∈ I .
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Polynomial Ring
x1, x2, . . . , xn coordinate functions on V = Cn
a polynomial f (x1, . . . , xn) can be viewed as function from V to CC[x] = C[x1, . . . , xn] graded ring of polynomial functions
Definition (Ideal)
a subset I ⊆ C[x] is an ideal if
1. 0 ∈ I ;
2. f (x), g(x) ∈ I ⇒ f (x) + g(x) ∈ I ;
3. f (x) ∈ C[x], g(x) ∈ I ⇒ f (x)g(x) ∈ I .
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Basis Theorem
the ideal (S) generated by a subset S ⊆ C[x] is
{a1(x)f1(x)+· · ·+ar (x)fr (x) | r ∈ N,∀i ai (x) ∈ C[x], fi (x) ∈ S}
Theorem (Hilbert 1890)
every ideal I ⊆ C[x] is generated by a finite set(C[x] is noetherian)
if S ⊆ C[x], then (S) = (T ) for some finite subset T ⊆ S
Hilbert used this theorem to prove a his Finiteness Theorem inInvariant Theory (discussed later)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Basis Theorem
the ideal (S) generated by a subset S ⊆ C[x] is
{a1(x)f1(x)+· · ·+ar (x)fr (x) | r ∈ N,∀i ai (x) ∈ C[x], fi (x) ∈ S}
Theorem (Hilbert 1890)
every ideal I ⊆ C[x] is generated by a finite set(C[x] is noetherian)
if S ⊆ C[x], then (S) = (T ) for some finite subset T ⊆ S
Hilbert used this theorem to prove a his Finiteness Theorem inInvariant Theory (discussed later)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Basis Theorem
the ideal (S) generated by a subset S ⊆ C[x] is
{a1(x)f1(x)+· · ·+ar (x)fr (x) | r ∈ N,∀i ai (x) ∈ C[x], fi (x) ∈ S}
Theorem (Hilbert 1890)
every ideal I ⊆ C[x] is generated by a finite set(C[x] is noetherian)
if S ⊆ C[x], then (S) = (T ) for some finite subset T ⊆ S
Hilbert used this theorem to prove a his Finiteness Theorem inInvariant Theory (discussed later)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Basis Theorem
the ideal (S) generated by a subset S ⊆ C[x] is
{a1(x)f1(x)+· · ·+ar (x)fr (x) | r ∈ N,∀i ai (x) ∈ C[x], fi (x) ∈ S}
Theorem (Hilbert 1890)
every ideal I ⊆ C[x] is generated by a finite set(C[x] is noetherian)
if S ⊆ C[x], then (S) = (T ) for some finite subset T ⊆ S
Hilbert used this theorem to prove a his Finiteness Theorem inInvariant Theory (discussed later)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Action of a Group G
suppose V = Cn is a representation of a group Gthis means that every g ∈ G acts my some n × n matrixMg : V → V (so g · v = Mgv)
and we have Me = I andMgh = MgMh
this also implies Mg−1 = (Mg )−1
if f (x) ∈ C[x] and M = (mi ,j) is n × n matrix, then v 7→ f (Mv) isa polynomial function given by the formula
f( n∑
j=1
m1,jxj , . . . ,n∑
j=1
mn,jxj
)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Action of a Group G
suppose V = Cn is a representation of a group Gthis means that every g ∈ G acts my some n × n matrixMg : V → V (so g · v = Mgv) and we have Me = I andMgh = MgMh
this also implies Mg−1 = (Mg )−1
if f (x) ∈ C[x] and M = (mi ,j) is n × n matrix, then v 7→ f (Mv) isa polynomial function given by the formula
f( n∑
j=1
m1,jxj , . . . ,n∑
j=1
mn,jxj
)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Action of a Group G
suppose V = Cn is a representation of a group Gthis means that every g ∈ G acts my some n × n matrixMg : V → V (so g · v = Mgv) and we have Me = I andMgh = MgMh
this also implies Mg−1 = (Mg )−1
if f (x) ∈ C[x] and M = (mi ,j) is n × n matrix, then v 7→ f (Mv) isa polynomial function given by the formula
f( n∑
j=1
m1,jxj , . . . ,n∑
j=1
mn,jxj
)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Action of a Group G
G acts on C[x] as follows:
if g ∈ G and f (x) ∈ C[x] then define (g · f )(x) ∈ C[x] by(g · f )(v) = f (Mg−1v)
(we use Mg−1 instead of Mg to make it a left action)
C[x] is an ∞-dimensional C-vector spacethe monomials form a basisG acts by linear transformations on C[x]C[x] is an ∞-dimensional representation of G
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Action of a Group G
G acts on C[x] as follows:
if g ∈ G and f (x) ∈ C[x] then define (g · f )(x) ∈ C[x] by(g · f )(v) = f (Mg−1v)
(we use Mg−1 instead of Mg to make it a left action)
C[x] is an ∞-dimensional C-vector spacethe monomials form a basisG acts by linear transformations on C[x]C[x] is an ∞-dimensional representation of G
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Invariant Ring
f (x) ∈ C[x] is G -invariant if (g · f )(x) = f (x) for all g ∈ Gf (x) ∈ C[x] is G -invariant if and only if it is constant on allG -orbits in V
Definition
C[x]G is the set of all G -invariant polynomials in C[x]
C[x]G is a subalgebra, i.e., contains C and is closed under addition,subtraction and multiplication
if f1(x), . . . , fr (x) ∈ C[x] then
C[f1(x), . . . , fr (x)] :=
{p(f1(x), . . . , fr (x)) | p(y1, . . . , yr ) ∈ C[y1, . . . , yr ]}
is the subalgebra of C[x] generated by f1(x), . . . , fr (x).
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Invariant Ring
f (x) ∈ C[x] is G -invariant if (g · f )(x) = f (x) for all g ∈ Gf (x) ∈ C[x] is G -invariant if and only if it is constant on allG -orbits in V
Definition
C[x]G is the set of all G -invariant polynomials in C[x]
C[x]G is a subalgebra, i.e., contains C and is closed under addition,subtraction and multiplication
if f1(x), . . . , fr (x) ∈ C[x] then
C[f1(x), . . . , fr (x)] :=
{p(f1(x), . . . , fr (x)) | p(y1, . . . , yr ) ∈ C[y1, . . . , yr ]}
is the subalgebra of C[x] generated by f1(x), . . . , fr (x).
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Invariant Ring
f (x) ∈ C[x] is G -invariant if (g · f )(x) = f (x) for all g ∈ Gf (x) ∈ C[x] is G -invariant if and only if it is constant on allG -orbits in V
Definition
C[x]G is the set of all G -invariant polynomials in C[x]
C[x]G is a subalgebra, i.e., contains C and is closed under addition,subtraction and multiplication
if f1(x), . . . , fr (x) ∈ C[x] then
C[f1(x), . . . , fr (x)] :=
{p(f1(x), . . . , fr (x)) | p(y1, . . . , yr ) ∈ C[y1, . . . , yr ]}
is the subalgebra of C[x] generated by f1(x), . . . , fr (x).
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Symmetric Group
G = Sn acts on V = Cn by permuting the coordinatesfor σ ∈ Sn, Mσ is the corresponding permutation matrixSn acts on C[x] as
(σ · f )(x1, . . . , xn) = f (xσ(1), . . . , xσ(n))
define the k-th elementary symmetric function as
ek(x) =∑
1≤i1<i2<···<ik≤nxi1xi2 · · · xik
for example e1 = x1 + x2 + · · ·+ xn and en = x1x2 · · · xn
Theorem
C[x]Sn = C[e1(x), . . . , en(x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Symmetric Group
G = Sn acts on V = Cn by permuting the coordinatesfor σ ∈ Sn, Mσ is the corresponding permutation matrixSn acts on C[x] as
(σ · f )(x1, . . . , xn) = f (xσ(1), . . . , xσ(n))
define the k-th elementary symmetric function as
ek(x) =∑
1≤i1<i2<···<ik≤nxi1xi2 · · · xik
for example e1 = x1 + x2 + · · ·+ xn and en = x1x2 · · · xn
Theorem
C[x]Sn = C[e1(x), . . . , en(x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
The Symmetric Group
G = Sn acts on V = Cn by permuting the coordinatesfor σ ∈ Sn, Mσ is the corresponding permutation matrixSn acts on C[x] as
(σ · f )(x1, . . . , xn) = f (xσ(1), . . . , xσ(n))
define the k-th elementary symmetric function as
ek(x) =∑
1≤i1<i2<···<ik≤nxi1xi2 · · · xik
for example e1 = x1 + x2 + · · ·+ xn and en = x1x2 · · · xn
Theorem
C[x]Sn = C[e1(x), . . . , en(x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Finiteness Theorem
assume that G is (linearly) reductive, which means that everyrepresentation of G is a direct sum of irreducible representationsexamples are GLn, SLn, On, finite groups
Theorem (Hilbert 1890)
C[x]G is a finitely generated algebra, i.e.,C[x]G = C[f1(x), . . . , fr (x)] for some r <∞ andf1(x), . . . , fr (x) ∈ C[x]G
proof sketch:J ⊆ C[x] ideal generated by all homogeneous, non-constantf (x) ∈ C[x]G (∞ many!)Basis Theorem: J = (f1(x), . . . , fr (x)) for some r <∞ andhomogeneous f1(x), . . . , fr (x) ∈ C[x]G
by induction one shows that C[x]G = C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Finiteness Theorem
assume that G is (linearly) reductive, which means that everyrepresentation of G is a direct sum of irreducible representationsexamples are GLn, SLn, On, finite groups
Theorem (Hilbert 1890)
C[x]G is a finitely generated algebra, i.e.,C[x]G = C[f1(x), . . . , fr (x)] for some r <∞ andf1(x), . . . , fr (x) ∈ C[x]G
proof sketch:J ⊆ C[x] ideal generated by all homogeneous, non-constantf (x) ∈ C[x]G (∞ many!)Basis Theorem: J = (f1(x), . . . , fr (x)) for some r <∞ andhomogeneous f1(x), . . . , fr (x) ∈ C[x]G
by induction one shows that C[x]G = C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Finiteness Theorem
assume that G is (linearly) reductive, which means that everyrepresentation of G is a direct sum of irreducible representationsexamples are GLn, SLn, On, finite groups
Theorem (Hilbert 1890)
C[x]G is a finitely generated algebra, i.e.,C[x]G = C[f1(x), . . . , fr (x)] for some r <∞ andf1(x), . . . , fr (x) ∈ C[x]G
proof sketch:J ⊆ C[x] ideal generated by all homogeneous, non-constantf (x) ∈ C[x]G (∞ many!)
Basis Theorem: J = (f1(x), . . . , fr (x)) for some r <∞ andhomogeneous f1(x), . . . , fr (x) ∈ C[x]G
by induction one shows that C[x]G = C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Finiteness Theorem
assume that G is (linearly) reductive, which means that everyrepresentation of G is a direct sum of irreducible representationsexamples are GLn, SLn, On, finite groups
Theorem (Hilbert 1890)
C[x]G is a finitely generated algebra, i.e.,C[x]G = C[f1(x), . . . , fr (x)] for some r <∞ andf1(x), . . . , fr (x) ∈ C[x]G
proof sketch:J ⊆ C[x] ideal generated by all homogeneous, non-constantf (x) ∈ C[x]G (∞ many!)Basis Theorem: J = (f1(x), . . . , fr (x)) for some r <∞ andhomogeneous f1(x), . . . , fr (x) ∈ C[x]G
by induction one shows that C[x]G = C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Finiteness Theorem
assume that G is (linearly) reductive, which means that everyrepresentation of G is a direct sum of irreducible representationsexamples are GLn, SLn, On, finite groups
Theorem (Hilbert 1890)
C[x]G is a finitely generated algebra, i.e.,C[x]G = C[f1(x), . . . , fr (x)] for some r <∞ andf1(x), . . . , fr (x) ∈ C[x]G
proof sketch:J ⊆ C[x] ideal generated by all homogeneous, non-constantf (x) ∈ C[x]G (∞ many!)Basis Theorem: J = (f1(x), . . . , fr (x)) for some r <∞ andhomogeneous f1(x), . . . , fr (x) ∈ C[x]G
by induction one shows that C[x]G = C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Definition
β(C[x]G ) is the smallest d such that C[x]G is generated bypolynomials of degree ≤ d
Theorem (Jordan 1876)
for binary forms of degree d we have β(C[x1, . . . , xd+1]SL2) ≤ d6
Theorem (Emmy Noether 1916)
if G is finite then β(C[x]G ) ≤ |G |
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Definition
β(C[x]G ) is the smallest d such that C[x]G is generated bypolynomials of degree ≤ d
Theorem (Jordan 1876)
for binary forms of degree d we have β(C[x1, . . . , xd+1]SL2) ≤ d6
Theorem (Emmy Noether 1916)
if G is finite then β(C[x]G ) ≤ |G |
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Definition
β(C[x]G ) is the smallest d such that C[x]G is generated bypolynomials of degree ≤ d
Theorem (Jordan 1876)
for binary forms of degree d we have β(C[x1, . . . , xd+1]SL2) ≤ d6
Theorem (Emmy Noether 1916)
if G is finite then β(C[x]G ) ≤ |G |
Harm Derksen, University of Michigan An Introduction to Invariant Theory
A Constructive Proof
the proof of Hilbert’s finiteness theorem does not give an algorithmfor finding generators, nor does it give an upper bound forβ(C[x]G ) for arbitrary G
so Hilbert gave another, more constructive proof in 1893 of hisFiniteness Theorem using his notion of the null cone
Harm Derksen, University of Michigan An Introduction to Invariant Theory
A Constructive Proof
the proof of Hilbert’s finiteness theorem does not give an algorithmfor finding generators, nor does it give an upper bound forβ(C[x]G ) for arbitrary G
so Hilbert gave another, more constructive proof in 1893 of hisFiniteness Theorem using his notion of the null cone
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Null cone
for v ∈ V , G · v = {g · v | g ∈ G} is orbit of vG · v ⊆ V closure of the orbit
Theorem
G · v ∩ G · w 6= ∅ ⇔ f (v) = f (w) for all f (x) ∈ C[x]G
⇒: f ∈ C[x]G is constant on G · v and G · w
Definition
Hilbert’s Null cone:
N := {v ∈ V | 0 ∈ G · v} =
= {v ∈ V | f (v) = f (0) for all f (x) ∈ C[x]G}
if C[x]G = C[f1(x), . . . , fr (x)] with f1(x), . . . , fr (x) homogeneous,non-constant, then N = {v ∈ V | f1(v) = · · · = fr (v) = 0}
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Null cone
for v ∈ V , G · v = {g · v | g ∈ G} is orbit of vG · v ⊆ V closure of the orbit
Theorem
G · v ∩ G · w 6= ∅ ⇔ f (v) = f (w) for all f (x) ∈ C[x]G
⇒: f ∈ C[x]G is constant on G · v and G · w
Definition
Hilbert’s Null cone:
N := {v ∈ V | 0 ∈ G · v} =
= {v ∈ V | f (v) = f (0) for all f (x) ∈ C[x]G}
if C[x]G = C[f1(x), . . . , fr (x)] with f1(x), . . . , fr (x) homogeneous,non-constant, then N = {v ∈ V | f1(v) = · · · = fr (v) = 0}
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert’s Null cone
for v ∈ V , G · v = {g · v | g ∈ G} is orbit of vG · v ⊆ V closure of the orbit
Theorem
G · v ∩ G · w 6= ∅ ⇔ f (v) = f (w) for all f (x) ∈ C[x]G
⇒: f ∈ C[x]G is constant on G · v and G · w
Definition
Hilbert’s Null cone:
N := {v ∈ V | 0 ∈ G · v} =
= {v ∈ V | f (v) = f (0) for all f (x) ∈ C[x]G}
if C[x]G = C[f1(x), . . . , fr (x)] with f1(x), . . . , fr (x) homogeneous,non-constant, then N = {v ∈ V | f1(v) = · · · = fr (v) = 0}
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Example: Multiplicative Group
G = C?, V = C4
for t ∈ C?, define
Mt =
t 0 0 00 t 0 00 0 t−1 00 0 0 t−1
t ·
v1v2v3v4
= Mt
v1v2v3v4
=
tv1tv2
t−1v3t−1v4
N = {v1 = v2 = 0} ∪ {v3 = v4 = 0}
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Example: Multiplicative Group
C[x1, x2, x3, x4]C?
= C[x1x3, x1x4, x2x3, x2x4]
N = {v1v3 = v1v4 = v2v3 = v2v4 = 0} = {v1 = v2 = 0}∪{v3 = v4 = 0}
Note that in this case, there is an algebraic relation between thegenerators, namely
(x1x3)(x2x4) = (x1x4)(x2x3)
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert-Mumford criterion
Definition
a one parameter subgroup (1-PSG) is a homomorphism ofalgebraic groups λ : C? → G
Theorem (Hilbert-Mumford criterion)
if v ∈ V = Cn, thenv ∈ N ⇔ there exists a 1-PSG λ : C? → G with lim
t→0λ(t) · v = 0
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Hilbert-Mumford criterion
Definition
a one parameter subgroup (1-PSG) is a homomorphism ofalgebraic groups λ : C? → G
Theorem (Hilbert-Mumford criterion)
if v ∈ V = Cn, thenv ∈ N ⇔ there exists a 1-PSG λ : C? → G with lim
t→0λ(t) · v = 0
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Conjugation of n × n Matrices
V = Matn,n, the space of n × n matricesG = GLn (the group of invertible n × n matrices) acts byconjugation: if A = (ai ,j) ∈ V and g ∈ G then g · A = gAg−1
if
(?) λ(t) =
tk1
. . .
tkn
with k1 ≥ k2 ≥ · · · ≥ kn, then
λ(t) · A = λ(t)Aλ(t)−1 = (tki−kjai ,j).
so limt→0 λ(t) · A = 0 if and only if A is strict upper triangular
every 1-PSG is of the form (?) after a base change, so
A ∈ N ⇔ A conjugate to strict upper triang. mat.⇔ A is nilpotent
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Conjugation of n × n Matrices
V = Matn,n, the space of n × n matricesG = GLn (the group of invertible n × n matrices) acts byconjugation: if A = (ai ,j) ∈ V and g ∈ G then g · A = gAg−1
if
(?) λ(t) =
tk1
. . .
tkn
with k1 ≥ k2 ≥ · · · ≥ kn, then
λ(t) · A = λ(t)Aλ(t)−1 = (tki−kjai ,j).
so limt→0 λ(t) · A = 0 if and only if A is strict upper triangular
every 1-PSG is of the form (?) after a base change, so
A ∈ N ⇔ A conjugate to strict upper triang. mat.⇔ A is nilpotent
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Conjugation of n × n Matrices
V = Matn,n, the space of n × n matricesG = GLn (the group of invertible n × n matrices) acts byconjugation: if A = (ai ,j) ∈ V and g ∈ G then g · A = gAg−1
if
(?) λ(t) =
tk1
. . .
tkn
with k1 ≥ k2 ≥ · · · ≥ kn, then
λ(t) · A = λ(t)Aλ(t)−1 = (tki−kjai ,j).
so limt→0 λ(t) · A = 0 if and only if A is strict upper triangular
every 1-PSG is of the form (?) after a base change, so
A ∈ N ⇔ A conjugate to strict upper triang. mat.⇔ A is nilpotent
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Conjugation of n × n Matrices
X = (xi ,j) where xi ,j are indeterminates
det(tI − X ) = tn − f1(x)tn−1 + · · ·+ (−1)nfn(x)
where x = x1,1, x1,2, . . . , xn,nf1(A) = trace(A), fn(A) = det(A)
Theorem
C[x]G = C[f1(x), . . . , fn(x)]
A ∈ N ⇔ f1(A) = · · · = fn(A) = 0⇔ det(tI−A) = tn ⇔ A nilpotent
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Conjugation of n × n Matrices
X = (xi ,j) where xi ,j are indeterminates
det(tI − X ) = tn − f1(x)tn−1 + · · ·+ (−1)nfn(x)
where x = x1,1, x1,2, . . . , xn,nf1(A) = trace(A), fn(A) = det(A)
Theorem
C[x]G = C[f1(x), . . . , fn(x)]
A ∈ N ⇔ f1(A) = · · · = fn(A) = 0⇔ det(tI−A) = tn ⇔ A nilpotent
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Theorem (Hilbert 1893)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous andN = {v | f1(v) = · · · = fr (v) = 0}
then there exists finitely many homogenous invariantsh1(x), . . . , hs(x) such that every invariant p(x) ∈ C[x]G is of theform
p(x) = a1(x)h1(x) + · · ·+ as(x)hs(x)
for some a1(x), . . . , as(x) ∈ C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Theorem (Hilbert 1893)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous andN = {v | f1(v) = · · · = fr (v) = 0}
then there exists finitely many homogenous invariantsh1(x), . . . , hs(x) such that every invariant p(x) ∈ C[x]G is of theform
p(x) = a1(x)h1(x) + · · ·+ as(x)hs(x)
for some a1(x), . . . , as(x) ∈ C[f1(x), . . . , fr (x)]
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Theorem (Popov 1980)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the samedegree d and N = {v | f1(v) = · · · = fr (v) = 0}
then there exists finitely many homogenous invariantsh1(x), . . . , hs(x) of degree at most n(d − 1) such that everyinvariant p(x) ∈ C[x]G is of the form
p(x) = a1(x)h1(x) + · · ·+ as(x)hs(x)
for some a1(x), . . . , as(x) ∈ C[f1(x), . . . , fr (x)]
in particular, β(C[x]G ) ≤ max{d , n(d − 1)} ≤ nd(because C[x]G = C[f1(x), . . . , fr (x), h1(x), . . . , hs(x)])
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Theorem (Popov 1980)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the samedegree d and N = {v | f1(v) = · · · = fr (v) = 0}
then there exists finitely many homogenous invariantsh1(x), . . . , hs(x) of degree at most n(d − 1) such that everyinvariant p(x) ∈ C[x]G is of the form
p(x) = a1(x)h1(x) + · · ·+ as(x)hs(x)
for some a1(x), . . . , as(x) ∈ C[f1(x), . . . , fr (x)]
in particular, β(C[x]G ) ≤ max{d , n(d − 1)} ≤ nd(because C[x]G = C[f1(x), . . . , fr (x), h1(x), . . . , hs(x)])
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Degree Bounds
Theorem (Popov 1980)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the samedegree d and N = {v | f1(v) = · · · = fr (v) = 0}
then there exists finitely many homogenous invariantsh1(x), . . . , hs(x) of degree at most n(d − 1) such that everyinvariant p(x) ∈ C[x]G is of the form
p(x) = a1(x)h1(x) + · · ·+ as(x)hs(x)
for some a1(x), . . . , as(x) ∈ C[f1(x), . . . , fr (x)]
in particular, β(C[x]G ) ≤ max{d , n(d − 1)} ≤ nd(because C[x]G = C[f1(x), . . . , fr (x), h1(x), . . . , hs(x)])
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polynomial Degree Bounds
Theorem (D.)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the degree atmost d and N = {v | f1(v) = · · · = fr (v) = 0}
then we haveβ(C[x]G ) ≤ 3
8nd2
for binary forms of degree n, the null cone is defined byhomogeneous invariants of degree ≤ 2n3, and we getβ(C[x]G ) ≤ 3
8(n + 1)(2n3)2 (which is slightly worse than Jordan’sbound)
if G is fixed then the bound β(C[x]G ) is polynomial in n (thedimension of V ) and the largest euclidean length among theweights appearing in the representation
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polynomial Degree Bounds
Theorem (D.)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the degree atmost d and N = {v | f1(v) = · · · = fr (v) = 0}
then we haveβ(C[x]G ) ≤ 3
8nd2
for binary forms of degree n, the null cone is defined byhomogeneous invariants of degree ≤ 2n3, and we getβ(C[x]G ) ≤ 3
8(n + 1)(2n3)2 (which is slightly worse than Jordan’sbound)
if G is fixed then the bound β(C[x]G ) is polynomial in n (thedimension of V ) and the largest euclidean length among theweights appearing in the representation
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polynomial Degree Bounds
Theorem (D.)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the degree atmost d and N = {v | f1(v) = · · · = fr (v) = 0}
then we haveβ(C[x]G ) ≤ 3
8nd2
for binary forms of degree n, the null cone is defined byhomogeneous invariants of degree ≤ 2n3, and we getβ(C[x]G ) ≤ 3
8(n + 1)(2n3)2 (which is slightly worse than Jordan’sbound)
if G is fixed then the bound β(C[x]G ) is polynomial in n (thedimension of V ) and the largest euclidean length among theweights appearing in the representation
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polynomial Degree Bounds
Theorem (D.)
suppose f1(x), . . . , fr (x) ∈ C[x]G are homogeneous of the degree atmost d and N = {v | f1(v) = · · · = fr (v) = 0}
then we haveβ(C[x]G ) ≤ 3
8nd2
for binary forms of degree n, the null cone is defined byhomogeneous invariants of degree ≤ 2n3, and we getβ(C[x]G ) ≤ 3
8(n + 1)(2n3)2 (which is slightly worse than Jordan’sbound)
if G is fixed then the bound β(C[x]G ) is polynomial in n (thedimension of V ) and the largest euclidean length among theweights appearing in the representation
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polarization
V representation of GC[x] = C[x1, . . . , xn] ring of polynomial functions on V = Cn
C[x, y] of polynomial functions on V ⊕ V = C2n
for f (x) ∈ C[x]G we can write
f (x1 + ty1, . . . , xn + tyn) = f0(x, y) + f1(x, y)t + · · ·+ fd(x, y)td
where f0(x, y), . . . , fd(x, y) ∈ C[x, y]G
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polarization
R[m] := C[x1,1, . . . , xn,1, . . . , x1,m, . . . , xn,m] is the ring ofpolynomial functions on Vm ∼= Matn,m
if m < s then we can polarize f (x) ∈ R[m]G to get invariants inR[s]G
Theorem (Weyl)
if s > n = dimV then polarizing generators from R[n]G givegenerators of R[s]G .
in particular, β(R[s]G ) = β(R[n]G )
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polarization
R[m] := C[x1,1, . . . , xn,1, . . . , x1,m, . . . , xn,m] is the ring ofpolynomial functions on Vm ∼= Matn,m
if m < s then we can polarize f (x) ∈ R[m]G to get invariants inR[s]G
Theorem (Weyl)
if s > n = dimV then polarizing generators from R[n]G givegenerators of R[s]G .
in particular, β(R[s]G ) = β(R[n]G )
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Polarization
R[m] := C[x1,1, . . . , xn,1, . . . , x1,m, . . . , xn,m] is the ring ofpolynomial functions on Vm ∼= Matn,m
if m < s then we can polarize f (x) ∈ R[m]G to get invariants inR[s]G
Theorem (Weyl)
if s > n = dimV then polarizing generators from R[n]G givegenerators of R[s]G .
in particular, β(R[s]G ) = β(R[n]G )
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Other Fields
instead of C, we can take any algebraically closed field ofcharacteristic 0
the degree bounds are valid for arbitrary fields of characteristic 0we need “algebraically closed” to make geometric statementsabout the null cone, orbits, etc.
most statements are either false, or more difficult to prove inpositive characteristic
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Other Fields
instead of C, we can take any algebraically closed field ofcharacteristic 0
the degree bounds are valid for arbitrary fields of characteristic 0we need “algebraically closed” to make geometric statementsabout the null cone, orbits, etc.
most statements are either false, or more difficult to prove inpositive characteristic
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Other Fields
instead of C, we can take any algebraically closed field ofcharacteristic 0
the degree bounds are valid for arbitrary fields of characteristic 0we need “algebraically closed” to make geometric statementsabout the null cone, orbits, etc.
most statements are either false, or more difficult to prove inpositive characteristic
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Other Fields
Weyl’s theorem is false in positive characteristic
Noether’s bound (β(C[x]G ) ≤ |G | for finite G ) is wrong in positivecharacteristic
invariant rings of reductive groups are also finitely generated inpositive characteristic, but the proof is harder (using theorems ofNagata and Haboush) and many of the geometric statementsabout the null cone etc. are still true
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Other Fields
Weyl’s theorem is false in positive characteristic
Noether’s bound (β(C[x]G ) ≤ |G | for finite G ) is wrong in positivecharacteristic
invariant rings of reductive groups are also finitely generated inpositive characteristic, but the proof is harder (using theorems ofNagata and Haboush) and many of the geometric statementsabout the null cone etc. are still true
Harm Derksen, University of Michigan An Introduction to Invariant Theory
Other Fields
Weyl’s theorem is false in positive characteristic
Noether’s bound (β(C[x]G ) ≤ |G | for finite G ) is wrong in positivecharacteristic
invariant rings of reductive groups are also finitely generated inpositive characteristic, but the proof is harder (using theorems ofNagata and Haboush) and many of the geometric statementsabout the null cone etc. are still true
Harm Derksen, University of Michigan An Introduction to Invariant Theory