An Introduction to Invariant Theory

Harm Derksen, University of Michigan

Optimization, Complexity and Invariant Theory Institute for Advanced Study, June 4, 2018

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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

Plan of the Talk

I applications of invariants

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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

Plan of the Talk

I applications of invariants I a classical, motivating example : binary forms

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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

Plan of the Talk

I applications of invariants I a classical, motivating example : binary forms I polynomial rings ideals

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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

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

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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

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

Harm Derksen, University of Michigan An Introduction to Invariant Theory I degree bounds for invariants I polarization of invariants and Weyl’s Theorem I Invariant Theory for other ﬁelds

Plan of the Talk

Harm Derksen, University of Michigan An Introduction to Invariant Theory I polarization of invariants and Weyl’s Theorem I Invariant Theory for other ﬁelds

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

Harm Derksen, University of Michigan An Introduction to Invariant Theory I Invariant Theory for other ﬁelds

Plan of the Talk

Harm Derksen, University of Michigan An Introduction to Invariant Theory Plan of the Talk

Harm Derksen, University of Michigan An Introduction to Invariant Theory the total energy in a physical system is an invariant as the system evolves over time

loop invariants can be used to prove the correctness of an algorithm although the number of iterations in a loop may vary, the loop invariant tell us to say something about the variables after the iterations

Applications of Invariants

Deﬁnition an invariant is a quantity or expression that stays the same under certain operations

Harm Derksen, University of Michigan An Introduction to Invariant Theory loop invariants can be used to prove the correctness of an algorithm although the number of iterations in a loop may vary, the loop invariant tell us to say something about the variables after the iterations

Applications of Invariants

Deﬁnition an invariant is a quantity or expression that stays the same under certain operations

the total energy in a physical system is an invariant as the system evolves over time

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 under certain operations

the total energy in a physical system is an invariant as the system evolves over time

Harm Derksen, University of Michigan An Introduction to Invariant Theory (co-)homology groups are invariants of topological manifolds

Applications of Invariants

Knot invariants (such as the Jones polynomial) can be used to distinguish knots

knot invariants remain unchanged under Reidemeister moves

Harm Derksen, University of Michigan An Introduction to Invariant Theory Applications of Invariants

Knot invariants (such as the Jones polynomial) can be used to distinguish 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 I invariants that remain unchanged under group symmetries such as rotations, permutations etc. we start with a motivating example from 19th century invariant theory

Invariant Theory

in invariant theory we restrict ourselves to I invariants that are polynomial functions on a vector space

Harm Derksen, University of Michigan An Introduction to Invariant Theory we start with a motivating example from 19th century 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 symmetries such as rotations, permutations etc.

Harm Derksen, University of Michigan An Introduction to Invariant Theory Invariant Theory

Harm Derksen, University of Michigan An Introduction to Invariant Theory a b SL = : ad − bc = 1 2 c d is the group of 2 × 2 matrices with determinant one 2 a matrix A ∈ SL2 gives a linear change of coordinates in C

the group SL2 acts on (the coeﬃcients of) binary forms: we make the substitution (z, w) 7→ (az + cw, bz + dw) and get another polynomial

0 0 2 0 0 2 p (z, w) = p(az + cw, bz + dw) = p1z + p2zw + p3w

Classical Invariant Theory: Binary Forms

a binary form of degree 2 is a polynomial 2 2 p(z, w) = p1z + p2zw + p3w

with p1, p2, p3 ∈ C

Harm Derksen, University of Michigan An Introduction to Invariant Theory the group SL2 acts on (the coeﬃcients of) binary forms: we make the substitution (z, w) 7→ (az + cw, bz + dw) and get another polynomial

0 0 2 0 0 2 p (z, w) = p(az + cw, bz + dw) = p1z + p2zw + p3w

Classical Invariant Theory: Binary Forms

a binary form of degree 2 is a polynomial 2 2 p(z, w) = p1z + p2zw + p3w

with p1, p2, p3 ∈ C

a b SL = : ad − bc = 1 2 c d is the group of 2 × 2 matrices with determinant one 2 a matrix A ∈ SL2 gives a linear change of coordinates in C

Harm Derksen, University of Michigan An Introduction to Invariant Theory Classical Invariant Theory: Binary Forms

a binary form of degree 2 is a polynomial 2 2 p(z, w) = p1z + p2zw + p3w

with p1, p2, p3 ∈ C

a b SL = : ad − bc = 1 2 c d is the group of 2 × 2 matrices with determinant one 2 a matrix A ∈ SL2 gives a linear change of coordinates in C

the group SL2 acts on (the coeﬃcients of) binary forms: we make the substitution (z, w) 7→ (az + cw, bz + dw) and get another polynomial

0 0 2 0 0 2 p (z, w) = p(az + cw, bz + dw) = p1z + p2zw + p3w

Harm Derksen, University of Michigan An Introduction to Invariant Theory Classical Invariant Theory: Binary Forms

where a b A = ∈ SL , c d 2  0    p1 p1 0 p2 = MA p2 0 p3 p3 and a2 ab b2  MA = ac ad + bc bd c2 cd d2

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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 a polynomial q(y) such that h(x1, x2, x3) = q(f (x1, x2, x3))

Classical Invariant Theory: Binary Forms

2 the polynomial f (x1, x2, x3) = x2 − 4x1x3 ∈ C[x1, x2, x3] (the 3 discriminant) can be viewed as a function from C to C and an easy calculation shows that

   0  p1 p1 2 0 2 0 0 0 f p2 = p2 − 4p1p3 = (p2) − 4p1p3 = f p2 0 p3 p3

Harm Derksen, University of Michigan An Introduction to Invariant Theory f (x1, x2, x3) is a fundamental invariant that generates all invariants: if h(x1, x2, x3) is another polynomial invariant, then there exists a polynomial q(y) such that h(x1, x2, x3) = q(f (x1, x2, x3))

Classical Invariant Theory: Binary Forms

2 the polynomial f (x1, x2, x3) = x2 − 4x1x3 ∈ C[x1, x2, x3] (the 3 discriminant) can be viewed as a function from C to C and an easy calculation shows that

   0  p1 p1 2 0 2 0 0 0 f p2 = p2 − 4p1p3 = (p2) − 4p1p3 = f p2 0 p3 p3

we say that f (x1, x2, x3) is an invariant under the action of SL2

Harm Derksen, University of Michigan An Introduction to Invariant Theory Classical Invariant Theory: Binary Forms

2 the polynomial f (x1, x2, x3) = x2 − 4x1x3 ∈ C[x1, x2, x3] (the 3 discriminant) can be viewed as a function from C to C and an easy calculation shows that

   0  p1 p1 2 0 2 0 0 0 f p2 = p2 − 4p1p3 = (p2) − 4p1p3 = f p2 0 p3 p3

we say that f (x1, x2, x3) is an invariant under the action of SL2

Harm Derksen, University of Michigan An Introduction to Invariant Theory the vector space of binary forms of degree n is an (n + 1)-dimensional representation of SL2

Classical Invariant Theory: Binary Forms

n+1 we may identify binary forms of degree n with vectors in C :   p1  p2  p zn + p zn−1w + ··· + p w n ↔   1 2 n+1  .   .  pn+1

Harm Derksen, University of Michigan An Introduction to Invariant Theory Classical Invariant Theory: Binary Forms

n+1 we may identify binary forms of degree n with vectors in C :   p1  p2  p zn + p zn−1w + ··· + p w n ↔   1 2 n+1  .   .  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 Theorem (Gordan 1868) for binary forms of degree d there exists a ﬁnite system of fundamental invariants that generate all invariants (i.e., every invariant is a polynomial expression in the fundamental invariants)

one of the main objectives was to ﬁnd an explicit system of fundamental invariants for binary forms up to degree d

(currently known for d ≤ 10)

Classical Invariant Theory: Binary Forms

polynomial invariants for binary forms of arbitrary degree were extensively studied in the 19th century by mathematicians like Boole, Sylvester, Cayley, Aronhold, Hermite, Eisenstein, Clebsch, Gordan, Lie, Klein, Capelli etc.

Harm Derksen, University of Michigan An Introduction to Invariant Theory one of the main objectives was to ﬁnd an explicit system of fundamental invariants for binary forms up to degree d

(currently known for d ≤ 10)

Classical Invariant Theory: Binary Forms

Theorem (Gordan 1868) for binary forms of degree d there exists a ﬁnite system of fundamental invariants that generate all invariants (i.e., every invariant is a polynomial expression in the fundamental invariants)

Harm Derksen, University of Michigan An Introduction to Invariant Theory Classical Invariant Theory: Binary Forms

one of the main objectives was to ﬁnd an explicit system of fundamental invariants for binary forms up to degree d

(currently known for d ≤ 10)

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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 .

The Polynomial Ring

n x1, x2,..., xn coordinate functions on V = C a polynomial f (x1,..., xn) can be viewed as function from V to C C[x] = C[x1,..., xn] graded ring of polynomial functions

Harm Derksen, University of Michigan An Introduction to Invariant Theory The Polynomial Ring

n x1, x2,..., xn coordinate functions on V = C a polynomial f (x1,..., xn) can be viewed as function from V to C C[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 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 in Invariant Theory (discussed later)

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}

Harm Derksen, University of Michigan An Introduction to Invariant Theory if S ⊆ C[x], then (S) = (T ) for some ﬁnite subset T ⊆ S

Hilbert used this theorem to prove a his Finiteness Theorem in Invariant Theory (discussed later)

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)

Harm Derksen, University of Michigan An Introduction to Invariant Theory Hilbert used this theorem to prove a his Finiteness Theorem in Invariant Theory (discussed later)

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

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 in Invariant Theory (discussed later)

Harm Derksen, University of Michigan An Introduction to Invariant Theory and we have Me = I and Mgh = Mg Mh −1 this also implies Mg −1 = (Mg )

if f (x) ∈ C[x] and M = (mi,j ) is n × n matrix, then v 7→ f (Mv) is a polynomial function given by the formula

n n X X f m1,j xj ,..., mn,j xj j=1 j=1

Action of a Group G

n suppose V = C is a representation of a group G this means that every g ∈ G acts my some n × n matrix Mg : V → V (so g · v = Mg v)

Harm Derksen, University of Michigan An Introduction to Invariant Theory if f (x) ∈ C[x] and M = (mi,j ) is n × n matrix, then v 7→ f (Mv) is a polynomial function given by the formula

n n X X f m1,j xj ,..., mn,j xj j=1 j=1

Action of a Group G

n suppose V = C is a representation of a group G this means that every g ∈ G acts my some n × n matrix Mg : V → V (so g · v = Mg v) and we have Me = I and Mgh = Mg Mh −1 this also implies Mg −1 = (Mg )

Harm Derksen, University of Michigan An Introduction to Invariant Theory Action of a Group G

if f (x) ∈ C[x] and M = (mi,j ) is n × n matrix, then v 7→ f (Mv) is a polynomial function given by the formula

n n X X f m1,j xj ,..., mn,j xj j=1 j=1

Harm Derksen, University of Michigan An Introduction to Invariant Theory C[x] is an ∞-dimensional C-vector space the monomials form a basis G acts by linear transformations on C[x] C[x] is an ∞-dimensional representation of G

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 −1 v)

(we use Mg −1 instead of Mg to make it a left action)

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 −1 v)

(we use Mg −1 instead of Mg to make it a left action)

C[x] is an ∞-dimensional C-vector space the monomials form a basis G 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 Deﬁnition G C[x] is the set of all G-invariant polynomials in C[x] G C[x] 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).

The Invariant Ring

f (x) ∈ C[x] is G-invariant if (g · f )(x) = f (x) for all g ∈ G f (x) ∈ C[x] is G-invariant if and only if it is constant on all G-orbits in V

Harm Derksen, University of Michigan An Introduction to Invariant Theory 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).

The Invariant Ring

f (x) ∈ C[x] is G-invariant if (g · f )(x) = f (x) for all g ∈ G f (x) ∈ C[x] is G-invariant if and only if it is constant on all G-orbits in V

Deﬁnition G C[x] is the set of all G-invariant polynomials in C[x] G C[x] is a subalgebra, i.e., contains C and is closed under addition, subtraction and multiplication

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 ∈ G f (x) ∈ C[x] is G-invariant if and only if it is constant on all G-orbits in V

Deﬁnition G C[x] is the set of all G-invariant polynomials in C[x] G C[x] 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 deﬁne the k-th elementary symmetric function as X ek (x) = xi1 xi2 ··· xik 1≤i1

for example e1 = x1 + x2 + ··· + xn and en = x1x2 ··· xn

Theorem

Sn C[x] = C[e1(x),..., en(x)]

The Symmetric Group

n G = Sn acts on V = C by permuting the coordinates for σ ∈ Sn, Mσ is the corresponding permutation matrix Sn acts on C[x] as

(σ · f )(x1,..., xn) = f (xσ(1),..., xσ(n))

Harm Derksen, University of Michigan An Introduction to Invariant Theory Theorem

Sn C[x] = C[e1(x),..., en(x)]

The Symmetric Group

n G = Sn acts on V = C by permuting the coordinates for σ ∈ Sn, Mσ is the corresponding permutation matrix Sn acts on C[x] as

(σ · f )(x1,..., xn) = f (xσ(1),..., xσ(n))

deﬁne the k-th elementary symmetric function as X ek (x) = xi1 xi2 ··· xik 1≤i1

for example e1 = x1 + x2 + ··· + xn and en = x1x2 ··· xn

Harm Derksen, University of Michigan An Introduction to Invariant Theory The Symmetric Group

n G = Sn acts on V = C by permuting the coordinates for σ ∈ Sn, Mσ is the corresponding permutation matrix Sn acts on C[x] as

(σ · f )(x1,..., xn) = f (xσ(1),..., xσ(n))

deﬁne the k-th elementary symmetric function as X ek (x) = xi1 xi2 ··· xik 1≤i1

for example e1 = x1 + x2 + ··· + xn and en = x1x2 ··· xn

Theorem

Sn C[x] = C[e1(x),..., en(x)]

Harm Derksen, University of Michigan An Introduction to Invariant Theory Theorem (Hilbert 1890) G C[x] is a ﬁnitely generated algebra, i.e., G C[x] = C[f1(x),..., fr (x)] for some r < ∞ and G f1(x),..., fr (x) ∈ C[x] proof sketch: J ⊆ C[x] ideal generated by all homogeneous, non-constant G f (x) ∈ C[x] (∞ many!) Basis Theorem: J = (f1(x),..., fr (x)) for some r < ∞ and G homogeneous f1(x),..., fr (x) ∈ C[x] G by induction one shows that C[x] = C[f1(x),..., fr (x)]

Hilbert’s Finiteness Theorem

assume that G is (linearly) reductive, which means that every representation of G is a direct sum of irreducible representations examples are GLn, SLn,On, ﬁnite groups

Harm Derksen, University of Michigan An Introduction to Invariant Theory proof sketch: J ⊆ C[x] ideal generated by all homogeneous, non-constant G f (x) ∈ C[x] (∞ many!) Basis Theorem: J = (f1(x),..., fr (x)) for some r < ∞ and G homogeneous f1(x),..., fr (x) ∈ C[x] G by induction one shows that C[x] = C[f1(x),..., fr (x)]

Hilbert’s Finiteness Theorem

Harm Derksen, University of Michigan An Introduction to Invariant Theory Basis Theorem: J = (f1(x),..., fr (x)) for some r < ∞ and G homogeneous f1(x),..., fr (x) ∈ C[x] G by induction one shows that C[x] = C[f1(x),..., fr (x)]

Hilbert’s Finiteness Theorem

assume that G is (linearly) reductive, which means that every representation of G is a direct sum of irreducible representations examples are GLn, SLn,On, ﬁnite groups Theorem (Hilbert 1890) G C[x] is a ﬁnitely generated algebra, i.e., G C[x] = C[f1(x),..., fr (x)] for some r < ∞ and G f1(x),..., fr (x) ∈ C[x] proof sketch: J ⊆ C[x] ideal generated by all homogeneous, non-constant G f (x) ∈ C[x] (∞ many!)

Harm Derksen, University of Michigan An Introduction to Invariant Theory G by induction one shows that C[x] = C[f1(x),..., fr (x)]

Hilbert’s Finiteness Theorem

Harm Derksen, University of Michigan An Introduction to Invariant Theory Hilbert’s Finiteness Theorem

Harm Derksen, University of Michigan An Introduction to Invariant Theory Theorem (Jordan 1876) SL 6 for binary forms of degree d we have β(C[x1,..., xd+1] 2 ) ≤ d

Theorem (Emmy Noether 1916) G if G is ﬁnite then β(C[x] ) ≤ |G|

Degree Bounds

Deﬁnition G G β(C[x] ) is the smallest d such that C[x] is generated by polynomials of degree ≤ d

Harm Derksen, University of Michigan An Introduction to Invariant Theory Theorem (Emmy Noether 1916) G if G is ﬁnite then β(C[x] ) ≤ |G|

Degree Bounds

Deﬁnition G G β(C[x] ) is the smallest d such that C[x] is generated by polynomials of degree ≤ d

Theorem (Jordan 1876) SL 6 for binary forms of degree d we have β(C[x1,..., xd+1] 2 ) ≤ d

Harm Derksen, University of Michigan An Introduction to Invariant Theory Degree Bounds

Deﬁnition G G β(C[x] ) is the smallest d such that C[x] is generated by polynomials of degree ≤ d

Theorem (Jordan 1876) SL 6 for binary forms of degree d we have β(C[x1,..., xd+1] 2 ) ≤ d

Theorem (Emmy Noether 1916) G if G is ﬁnite then β(C[x] ) ≤ |G|

Harm Derksen, University of Michigan An Introduction to Invariant Theory so Hilbert gave another, more constructive proof in 1893 of his Finiteness Theorem using his notion of the null cone

A Constructive Proof

the proof of Hilbert’s ﬁniteness theorem does not give an algorithm for ﬁnding generators, nor does it give an upper bound for G β(C[x] ) for arbitrary 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 algorithm for ﬁnding generators, nor does it give an upper bound for G β(C[x] ) for arbitrary G so Hilbert gave another, more constructive proof in 1893 of his Finiteness Theorem using his notion of the null cone

Harm Derksen, University of Michigan An Introduction to Invariant Theory G ⇒: f ∈ C[x] is constant on G · v and G · w Deﬁnition Hilbert’s Null cone:

N := {v ∈ V | 0 ∈ G · v} = G = {v ∈ V | f (v) = f (0) for all f (x) ∈ C[x] }

G if C[x] = C[f1(x),..., fr (x)] with f1(x),..., fr (x) homogeneous, non-constant, then N = {v ∈ V | f1(v) = ··· = fr (v) = 0}

Hilbert’s Null cone

for v ∈ V , G · v = {g · v | g ∈ G} is orbit of v G · v ⊆ V closure of the orbit Theorem G G · v ∩ G · w 6= ∅ ⇔ f (v) = f (w) for all f (x) ∈ C[x]

Harm Derksen, University of Michigan An Introduction to Invariant Theory Deﬁnition Hilbert’s Null cone:

N := {v ∈ V | 0 ∈ G · v} = G = {v ∈ V | f (v) = f (0) for all f (x) ∈ C[x] }

G if C[x] = C[f1(x),..., fr (x)] with f1(x),..., fr (x) homogeneous, non-constant, then N = {v ∈ V | f1(v) = ··· = fr (v) = 0}

Hilbert’s Null cone

for v ∈ V , G · v = {g · v | g ∈ G} is orbit of v G · v ⊆ V closure of the orbit Theorem G G · v ∩ G · w 6= ∅ ⇔ f (v) = f (w) for all f (x) ∈ C[x] G ⇒: f ∈ C[x] is constant on G · v and G · w

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 v G · v ⊆ V closure of the orbit Theorem G G · v ∩ G · w 6= ∅ ⇔ f (v) = f (w) for all f (x) ∈ C[x] G ⇒: f ∈ C[x] is constant on G · v and G · w Deﬁnition Hilbert’s Null cone:

N := {v ∈ V | 0 ∈ G · v} = G = {v ∈ V | f (v) = f (0) for all f (x) ∈ C[x] }

G if C[x] = 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

? 4 G = C , V = C ? for t ∈ C , deﬁne t 0 0 0  0 t 0 0  Mt =   0 0 t−1 0  0 0 0 t−1       v1 v1 tv1 v2 v2  tv2  t ·   = Mt   =  −1  v3 v3 t v3 −1 v4 v4 t v4

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 the generators, namely

(x1x3)(x2x4) = (x1x4)(x2x3)

Harm Derksen, University of Michigan An Introduction to Invariant Theory Theorem (Hilbert-Mumford criterion) n if v ∈ V = C , then ? v ∈ N ⇔ there exists a 1-PSG λ : C → G with lim λ(t) · v = 0 t→0

Hilbert-Mumford criterion

Deﬁnition a one parameter subgroup (1-PSG) is a homomorphism of ? algebraic groups λ : C → G

Harm Derksen, University of Michigan An Introduction to Invariant Theory Hilbert-Mumford criterion

Deﬁnition a one parameter subgroup (1-PSG) is a homomorphism of ? algebraic groups λ : C → G

Theorem (Hilbert-Mumford criterion) n if v ∈ V = C , then ? v ∈ N ⇔ there exists a 1-PSG λ : C → G with lim λ(t) · v = 0 t→0

Harm Derksen, University of Michigan An Introduction to Invariant Theory if tk1   ..  (?) λ(t) =  .  tkn

with k1 ≥ k2 ≥ · · · ≥ kn, then

−1 ki −kj λ(t) · A = λ(t)Aλ(t) = (t ai,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

Conjugation of n × n Matrices

V = Matn,n, the space of n × n matrices G = GLn (the group of invertible n × n matrices) acts by −1 conjugation: if A = (ai,j ) ∈ V and g ∈ G then g · A = gAg

Harm Derksen, University of Michigan An Introduction to Invariant Theory every 1-PSG is of the form (?) after a base change, so

A ∈ N ⇔ A conjugate to strict upper triang. mat. ⇔ A is nilpotent

Conjugation of n × n Matrices

V = Matn,n, the space of n × n matrices G = GLn (the group of invertible n × n matrices) acts by −1 conjugation: if A = (ai,j ) ∈ V and g ∈ G then g · A = gAg

if tk1   ..  (?) λ(t) =  .  tkn

with k1 ≥ k2 ≥ · · · ≥ kn, then

−1 ki −kj λ(t) · A = λ(t)Aλ(t) = (t ai,j ).

so limt→0 λ(t) · A = 0 if and only if A is strict upper triangular

Harm Derksen, University of Michigan An Introduction to Invariant Theory Conjugation of n × n Matrices

V = Matn,n, the space of n × n matrices G = GLn (the group of invertible n × n matrices) acts by −1 conjugation: if A = (ai,j ) ∈ V and g ∈ G then g · A = gAg

if tk1   ..  (?) λ(t) =  .  tkn

with k1 ≥ k2 ≥ · · · ≥ kn, then

−1 ki −kj λ(t) · A = λ(t)Aλ(t) = (t ai,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 Theorem G C[x] = C[f1(x),..., fn(x)]

n A ∈ N ⇔ f1(A) = ··· = fn(A) = 0 ⇔ det(tI −A) = t ⇔ A nilpotent

Conjugation of n × n Matrices

X = (xi,j ) where xi,j are indeterminates

n n−1 n det(tI − X ) = t − f1(x)t + ··· + (−1) fn(x)

where x = x1,1, x1,2,..., xn,n f1(A) = trace(A), fn(A) = det(A)

Harm Derksen, University of Michigan An Introduction to Invariant Theory Conjugation of n × n Matrices

X = (xi,j ) where xi,j are indeterminates

n n−1 n det(tI − X ) = t − f1(x)t + ··· + (−1) fn(x)

where x = x1,1, x1,2,..., xn,n f1(A) = trace(A), fn(A) = det(A) Theorem G C[x] = C[f1(x),..., fn(x)]

n A ∈ N ⇔ f1(A) = ··· = fn(A) = 0 ⇔ det(tI −A) = t ⇔ A nilpotent

Harm Derksen, University of Michigan An Introduction to Invariant Theory then there exists ﬁnitely many homogenous invariants G h1(x),..., hs (x) such that every invariant p(x) ∈ C[x] 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)]

Degree Bounds

Theorem (Hilbert 1893) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous and N = {v | f1(v) = ··· = fr (v) = 0}

Harm Derksen, University of Michigan An Introduction to Invariant Theory Degree Bounds

Theorem (Hilbert 1893) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous and N = {v | f1(v) = ··· = fr (v) = 0}

then there exists ﬁnitely many homogenous invariants G h1(x),..., hs (x) such that every invariant p(x) ∈ C[x] 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)]

Harm Derksen, University of Michigan An Introduction to Invariant Theory then there exists ﬁnitely many homogenous invariants h1(x),..., hs (x) of degree at most n(d − 1) such that every G invariant p(x) ∈ C[x] 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)]

G in particular, β(C[x] ) ≤ max{d, n(d − 1)} ≤ nd G (because C[x] = C[f1(x),..., fr (x), h1(x),..., hs (x)])

Degree Bounds

Theorem (Popov 1980) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the same degree d and N = {v | f1(v) = ··· = fr (v) = 0}

Harm Derksen, University of Michigan An Introduction to Invariant Theory G in particular, β(C[x] ) ≤ max{d, n(d − 1)} ≤ nd G (because C[x] = C[f1(x),..., fr (x), h1(x),..., hs (x)])

Degree Bounds

Theorem (Popov 1980) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the same degree d and N = {v | f1(v) = ··· = fr (v) = 0}

then there exists ﬁnitely many homogenous invariants h1(x),..., hs (x) of degree at most n(d − 1) such that every G invariant p(x) ∈ C[x] 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)]

Harm Derksen, University of Michigan An Introduction to Invariant Theory Degree Bounds

Theorem (Popov 1980) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the same degree d and N = {v | f1(v) = ··· = fr (v) = 0}

then there exists ﬁnitely many homogenous invariants h1(x),..., hs (x) of degree at most n(d − 1) such that every G invariant p(x) ∈ C[x] 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)]

G in particular, β(C[x] ) ≤ max{d, n(d − 1)} ≤ nd G (because C[x] = C[f1(x),..., fr (x), h1(x),..., hs (x)])

Harm Derksen, University of Michigan An Introduction to Invariant Theory then we have G 3 2 β(C[x] ) ≤ 8 nd

for binary forms of degree n, the null cone is deﬁned by homogeneous invariants of degree ≤ 2n3, and we get G 3 3 2 β(C[x] ) ≤ 8 (n + 1)(2n ) (which is slightly worse than Jordan’s bound)

G if G is ﬁxed then the bound β(C[x] ) is polynomial in n (the dimension of V ) and the largest euclidean length among the weights appearing in the representation

Polynomial Degree Bounds

Theorem (D.) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the degree at most d and N = {v | f1(v) = ··· = fr (v) = 0}

Harm Derksen, University of Michigan An Introduction to Invariant Theory for binary forms of degree n, the null cone is deﬁned by homogeneous invariants of degree ≤ 2n3, and we get G 3 3 2 β(C[x] ) ≤ 8 (n + 1)(2n ) (which is slightly worse than Jordan’s bound)

G if G is ﬁxed then the bound β(C[x] ) is polynomial in n (the dimension of V ) and the largest euclidean length among the weights appearing in the representation

Polynomial Degree Bounds

Theorem (D.) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the degree at most d and N = {v | f1(v) = ··· = fr (v) = 0}

then we have G 3 2 β(C[x] ) ≤ 8 nd

Harm Derksen, University of Michigan An Introduction to Invariant Theory G if G is ﬁxed then the bound β(C[x] ) is polynomial in n (the dimension of V ) and the largest euclidean length among the weights appearing in the representation

Polynomial Degree Bounds

Theorem (D.) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the degree at most d and N = {v | f1(v) = ··· = fr (v) = 0}

then we have G 3 2 β(C[x] ) ≤ 8 nd

for binary forms of degree n, the null cone is deﬁned by homogeneous invariants of degree ≤ 2n3, and we get G 3 3 2 β(C[x] ) ≤ 8 (n + 1)(2n ) (which is slightly worse than Jordan’s bound)

Harm Derksen, University of Michigan An Introduction to Invariant Theory Polynomial Degree Bounds

Theorem (D.) G suppose f1(x),..., fr (x) ∈ C[x] are homogeneous of the degree at most d and N = {v | f1(v) = ··· = fr (v) = 0}

then we have G 3 2 β(C[x] ) ≤ 8 nd

for binary forms of degree n, the null cone is deﬁned by homogeneous invariants of degree ≤ 2n3, and we get G 3 3 2 β(C[x] ) ≤ 8 (n + 1)(2n ) (which is slightly worse than Jordan’s bound)

G if G is ﬁxed then the bound β(C[x] ) is polynomial in n (the dimension of V ) and the largest euclidean length among the weights appearing in the representation

Harm Derksen, University of Michigan An Introduction to Invariant Theory Polarization

V representation of G n C[x] = C[x1,..., xn] ring of polynomial functions on V = C 2n C[x, y] of polynomial functions on V ⊕ V = C

G for f (x) ∈ C[x] we can write

d f (x1 + ty1,..., xn + tyn) = f0(x, y) + f1(x, y)t + ··· + fd (x, y)t

G where f0(x, y),..., fd (x, y) ∈ C[x, y]

Harm Derksen, University of Michigan An Introduction to Invariant Theory if m < s then we can polarize f (x) ∈ R[m]G to get invariants in R[s]G Theorem (Weyl) if s > n = dim V then polarizing generators from R[n]G give generators of R[s]G .

in particular, β(R[s]G ) = β(R[n]G )

Polarization

R[m] := C[x1,1,..., xn,1,..., x1,m,..., xn,m] is the ring of m ∼ polynomial functions on V = Matn,m

Harm Derksen, University of Michigan An Introduction to Invariant Theory in particular, β(R[s]G ) = β(R[n]G )

Polarization

R[m] := C[x1,1,..., xn,1,..., x1,m,..., xn,m] is the ring of m ∼ polynomial functions on V = Matn,m

if m < s then we can polarize f (x) ∈ R[m]G to get invariants in R[s]G Theorem (Weyl) if s > n = dim V then polarizing generators from R[n]G give generators of R[s]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 of m ∼ polynomial functions on V = Matn,m

if m < s then we can polarize f (x) ∈ R[m]G to get invariants in R[s]G Theorem (Weyl) if s > n = dim V then polarizing generators from R[n]G give generators of R[s]G .

in particular, β(R[s]G ) = β(R[n]G )

Harm Derksen, University of Michigan An Introduction to Invariant Theory the degree bounds are valid for arbitrary ﬁelds of characteristic 0 we need “algebraically closed” to make geometric statements about the null cone, orbits, etc.

most statements are either false, or more diﬃcult to prove in positive characteristic

Other Fields

instead of C, we can take any algebraically closed ﬁeld of characteristic 0

Harm Derksen, University of Michigan An Introduction to Invariant Theory most statements are either false, or more diﬃcult to prove in positive characteristic

Other Fields

instead of C, we can take any algebraically closed ﬁeld of characteristic 0

the degree bounds are valid for arbitrary ﬁelds of characteristic 0 we need “algebraically closed” to make geometric statements about the null cone, orbits, etc.

Harm Derksen, University of Michigan An Introduction to Invariant Theory Other Fields

instead of C, we can take any algebraically closed ﬁeld of characteristic 0

the degree bounds are valid for arbitrary ﬁelds of characteristic 0 we need “algebraically closed” to make geometric statements about the null cone, orbits, etc.

most statements are either false, or more diﬃcult to prove in positive characteristic

Harm Derksen, University of Michigan An Introduction to Invariant Theory G Noether’s bound (β(C[x] ) ≤ |G| for ﬁnite G) is wrong in positive characteristic

invariant rings of reductive groups are also ﬁnitely generated in positive characteristic, but the proof is harder (using theorems of Nagata and Haboush) and many of the geometric statements about the null cone etc. are still true

Other Fields

Weyl’s theorem is false in positive characteristic

Harm Derksen, University of Michigan An Introduction to Invariant Theory invariant rings of reductive groups are also ﬁnitely generated in positive characteristic, but the proof is harder (using theorems of Nagata and Haboush) and many of the geometric statements about the null cone etc. are still true

Other Fields

Weyl’s theorem is false in positive characteristic

G Noether’s bound (β(C[x] ) ≤ |G| for ﬁnite G) is wrong in positive characteristic

Harm Derksen, University of Michigan An Introduction to Invariant Theory Other Fields

Weyl’s theorem is false in positive characteristic

G Noether’s bound (β(C[x] ) ≤ |G| for ﬁnite G) is wrong in positive characteristic

Harm Derksen, University of Michigan An Introduction to Invariant Theory