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Tensor Invariants and Kronecker Coefficients

Jiarui Fei

University of California, Riverside

November 23, 2013

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Tensor Invariants

By a (tri-)tensor of vector spaces (U, V , W ), we mean the vector space U∗ ⊗ V ⊗ W ∗. The product of special linear groups G := SL(U) × SL(V ) × SL(W ) acts naturally on it. We are interested in the ring of invariants k[U∗ ⊗ V ⊗ W ∗]G = Sym(U ⊗ V ∗ ⊗ W )G. The vector space can be identified with the (n1, n2)-dimensional representation space of the m-arrow Kronecker quiver Km, where dim U = n1, dim V = n2, and dim W = m. The ring of invariants SI(n1,n2)(Km) := k[U∗ ⊗ V ⊗ W ∗]SL(U)×SL(V ) is generated by Schofield’s determinantal invariants (Derksen-Weyman, Schofield-Van den Bergh).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Tensor Invariants

By a (tri-)tensor of vector spaces (U, V , W ), we mean the vector space U∗ ⊗ V ⊗ W ∗. The product of special linear groups G := SL(U) × SL(V ) × SL(W ) acts naturally on it. We are interested in the ring of invariants k[U∗ ⊗ V ⊗ W ∗]G = Sym(U ⊗ V ∗ ⊗ W )G. The vector space can be identified with the (n1, n2)-dimensional representation space of the m-arrow Kronecker quiver Km, where dim U = n1, dim V = n2, and dim W = m. The ring of invariants SI(n1,n2)(Km) := k[U∗ ⊗ V ⊗ W ∗]SL(U)×SL(V ) is generated by Schofield’s determinantal invariants (Derksen-Weyman, Schofield-Van den Bergh).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Variation of Schofield’s Invariants

For simplicity, we will assume n1 = n2 = n. By a semi-invariants of degree d, we mean an element f ∈ SI(n,n)(Km) of weight (d, −d), i.e., for any x ∈ U∗ ⊗ V ⊗ W ∗ and (g1, g2) ∈ GL(U) × GL(V ) we have that f ((g1, g2)x) = det(g1)d det(g2)−df (x). We consider the determinant of the dn × dn matrix

m

• k=1

Λk ⊗ Ak, where Λk = (λk

ij) is a d × d matrix and ⊗ is the Kronecker product.

Let c(D1, D2, . . . , Dm) be the coefficient of

i,j,k(λk ij)(Dk)ij in the

expansion.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Variation of Schofield’s Invariants

For simplicity, we will assume n1 = n2 = n. By a semi-invariants of degree d, we mean an element f ∈ SI(n,n)(Km) of weight (d, −d), i.e., for any x ∈ U∗ ⊗ V ⊗ W ∗ and (g1, g2) ∈ GL(U) × GL(V ) we have that f ((g1, g2)x) = det(g1)d det(g2)−df (x). We consider the determinant of the dn × dn matrix

m

• k=1

Λk ⊗ Ak, where Λk = (λk

ij) is a d × d matrix and ⊗ is the Kronecker product.

Let c(D1, D2, . . . , Dm) be the coefficient of

i,j,k(λk ij)(Dk)ij in the

expansion.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Example (m = 2, d = 3)

  λ1

11A1 + λ2 11A2

λ1

12A1 + λ2 12A2

λ1

13A1 + λ2 13A2

λ1

21A1 + λ2 21A2

λ1

22A1 + λ2 22A2

λ1

23A1 + λ2 23A2

λ1

31A1 + λ2 31A2

λ1

32A1 + λ2 32A2

λ1

33A1 + λ2 33A2

 

Lemma

{c(D1, D2, . . . , Dm)} linearly span the quiver invariants SId

(n,n)(Km).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Example (m = 2, d = 3)

  λ1

11A1 + λ2 11A2

λ1

12A1 + λ2 12A2

λ1

13A1 + λ2 13A2

λ1

21A1 + λ2 21A2

λ1

22A1 + λ2 22A2

λ1

23A1 + λ2 23A2

λ1

31A1 + λ2 31A2

λ1

32A1 + λ2 32A2

λ1

33A1 + λ2 33A2

 

Lemma

{c(D1, D2, . . . , Dm)} linearly span the quiver invariants SId

(n,n)(Km).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Relations

We view c(D1, D2, . . . , Dm) as formal variables, and denote F(C, Λ) :=

• c(D1, D2, . . . , Dm)
• i,j,k

(λk

ij)(Dk)ij.

Since we have for any elementary matrix Fij = I + Eij that, det(

• k

FijΛk ⊗ Ak) = det(

• k

Λk ⊗ Ak), such an equality generates a set of relations among c(D1, D2, . . . , Dm), namely F(C, FijΛ) − F(C, Λ) = 0.

Theorem

The relations below span all relations among c(D1, D2, . . . , Dm)’s. F(C, FijΛ) − F(C, Λ) = 0, and F(C, ΛFij) − F(C, Λ) = 0.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Relations

We view c(D1, D2, . . . , Dm) as formal variables, and denote F(C, Λ) :=

• c(D1, D2, . . . , Dm)
• i,j,k

(λk

ij)(Dk)ij.

Since we have for any elementary matrix Fij = I + Eij that, det(

• k

FijΛk ⊗ Ak) = det(

• k

Λk ⊗ Ak), such an equality generates a set of relations among c(D1, D2, . . . , Dm), namely F(C, FijΛ) − F(C, Λ) = 0.

Theorem

The relations below span all relations among c(D1, D2, . . . , Dm)’s. F(C, FijΛ) − F(C, Λ) = 0, and F(C, ΛFij) − F(C, Λ) = 0.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

The Lie Algebra Action

This set of generators behave well under the Lie algebra sl(W )

• action. Let ∂kl be the differential operator corresponding to the

matrix Ekl ∈ sl(W ). Then we have the following formula ∂kl

• c(D1, . . . , Dm)
• =
• (Dk)ij>0

−

• (Dl)ij+1
• c(. . . , Dk−Eij, . . . , Dl+Eij, . . . )

In particular, c(D) is T-invariant iff. m|dn and

n

• i=1

n

• j=1

Dij = dn m (1, 1, . . . , 1). (In priori,

n

• i=1

m

• k=1

Dk =

n

• j=1

m

• k=1

Dk = n(1, 1, . . . , 1).)

Jiarui Fei Tensor Invariants and Kronecker Coefficients

The Lie Algebra Action

This set of generators behave well under the Lie algebra sl(W )

• action. Let ∂kl be the differential operator corresponding to the

matrix Ekl ∈ sl(W ). Then we have the following formula ∂kl

• c(D1, . . . , Dm)
• =
• (Dk)ij>0

−

• (Dl)ij+1
• c(. . . , Dk−Eij, . . . , Dl+Eij, . . . )

In particular, c(D) is T-invariant iff. m|dn and

n

• i=1

n

• j=1

Dij = dn m (1, 1, . . . , 1). (In priori,

n

• i=1

m

• k=1

Dk =

n

• j=1

m

• k=1

Dk = n(1, 1, . . . , 1).)

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Finite-type Cases

222 The degree 2 invariant is Cayley’s hyperdeterminant c(11)2 − 4c(20)c(02). 233 The degree 4 invariant is c(12)2c(21)2 − 4c(03)c(21)3 − 4c(12)3c(30) +18c(03)c(12)c(21)c(30) − 27c(03)2c(30)2. 223 The degree 3 invariant is c 000 001 100

001 000 010 010 100 000

• − c

000 001 100

100 000 010 010 001 000

• = det
• A3

−A2 −A3 A1 A2 −A1

• .

224 The degree 2 invariant is c( 1000 0100

0010 0001 ) − c( 1000 0010 0100 0001 ),

whose square is the classical invariant of quaternary quadratic

• form. The invariant ring SI(2,2)(K4) is free over Sym(S2) with

basis S(14), and has equivariant free resolution S(24) ⊗ A(−4) → A,

Jiarui Fei Tensor Invariants and Kronecker Coefficients

The 333 Case

Proposition

The tensor invariants are generated by three algebraically independent elements. The degree 2 invariant is c( 111 000

000 111 ) + 4

• c( 020 100

100 002 ) + c( 002 010 010 200 ) + c( 200 001 001 020 )

• .

The degree 3 invariant is det

• A3

−A2 −A3 A1 A2 −A1

• .

The degree 4 invariant is the Aronhold invariant of ternary cubic form. The invariant ring SI(3,3)(K3) is free over Sym(S3 ⊕ S(23)) with basis S(33), and has equivariant free resolution S(63) ⊗ A(−6) → A, where A = Sym(S3 ⊕ S(23) ⊕ S(33)).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

The 333 Case

Proposition

The tensor invariants are generated by three algebraically independent elements. The degree 2 invariant is c( 111 000

000 111 ) + 4

• c( 020 100

100 002 ) + c( 002 010 010 200 ) + c( 200 001 001 020 )

• .

The degree 3 invariant is det

• A3

−A2 −A3 A1 A2 −A1

• .

The degree 4 invariant is the Aronhold invariant of ternary cubic form. The invariant ring SI(3,3)(K3) is free over Sym(S3 ⊕ S(23)) with basis S(33), and has equivariant free resolution S(63) ⊗ A(−6) → A, where A = Sym(S3 ⊕ S(23) ⊕ S(33)).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Kronecker Coefficients

We want to know in advance how many linearly independent invariants in each degree and the degree-bounds. The number of invariants are given by some special Kronecker coefficients. Kronecker coefficients are by definition the structure coefficient of the representation ring of the symmetric groups SµSν =

• λ

kλ

µ,νSλ.

By Schur-Weyl duality, it also appears as Sλ(V ⊗ W ) ∼ =

• µ,ν

kλ

µ,νSµ(V ) ⊗ Sν(W ).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Kronecker Coefficients from Quivers

Consider the following quiver K p,q

m

−p

· · · −2 −1

m 1

2 · · · q

Let the weight σ = (σ−p, . . . , σ−1, −σ1, . . . , −σq), and µT = (ασ−1

−1 , . . . , ασ−p −p , ), νT = (ασ1 1 , . . . , ασq q ) with |µ| = |ν| = n.

Then standard Schubert calculus shows that SIσ

α(K p,q m ) =

• |λ|=n

kλ

µ,νSλW

In particular, kλ

µ,ν = dim SIσ α(K p,q m )U(W ) λ

.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Kronecker Coefficients from Quivers

Consider the following quiver K p,q

m

−p

· · · −2 −1

m 1

2 · · · q

Let the weight σ = (σ−p, . . . , σ−1, −σ1, . . . , −σq), and µT = (ασ−1

−1 , . . . , ασ−p −p , ), νT = (ασ1 1 , . . . , ασq q ) with |µ| = |ν| = n.

Then standard Schubert calculus shows that SIσ

α(K p,q m ) =

• |λ|=n

kλ

µ,νSλW

In particular, kλ

µ,ν = dim SIσ α(K p,q m )U(W ) λ

.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

On the other hand, dim SIσ

α(K p,q m ) =

• cµ

η1,η2,...,ηmcν η1,η2,...,ηm

Assume λ has m parts. By Cauchy and Jacobi-Trudi formula, kλ

µ,ν =

• ω∈Sm

sgn(ω)

• |ηi|=λi−i+ω(i)

cµ

η1,η2,...,ηmcν η1,η2,...,ηm.

Each component

• |ηi|=λi−i+ω(i)

cµ

η1,η2,...,ηmcν η1,η2,...,ηm is the dimension

• f a T-weight space of SIσ

α(K p,q m ). This leads to an algorithm.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

On the other hand, dim SIσ

α(K p,q m ) =

• cµ

η1,η2,...,ηmcν η1,η2,...,ηm

Assume λ has m parts. By Cauchy and Jacobi-Trudi formula, kλ

µ,ν =

• ω∈Sm

sgn(ω)

• |ηi|=λi−i+ω(i)

cµ

η1,η2,...,ηmcν η1,η2,...,ηm.

Each component

• |ηi|=λi−i+ω(i)

cµ

η1,η2,...,ηmcν η1,η2,...,ηm is the dimension

• f a T-weight space of SIσ

α(K p,q m ). This leads to an algorithm.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Some Trivial Facts

• 1. It follows from the quiver invariant construction that for fixed

λ, µ, ν, the stretched Kronecker coefficients kcλ

cµ,cν is a

quasi-polynomial in c, but not a polynomial in general. It does not have the saturation property even in certain weak sense.

• 2. Let µk be the reflection at a sink/source k, then we have

isomorphism of invariant rings SIσ

α(Q) ∼

= SIµk(σ)

µk(α)(µk(Q)).

(1) So we obtain many equalities among Kronecker coefficients. (Question: how to find these equalities conceptually?) Moreover, the tensor invariants of U × V × W and U × V ′ × W are the same if dim U × dim W − dim V = dim V ′.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Some Trivial Facts

• 1. It follows from the quiver invariant construction that for fixed

λ, µ, ν, the stretched Kronecker coefficients kcλ

cµ,cν is a

quasi-polynomial in c, but not a polynomial in general. It does not have the saturation property even in certain weak sense.

• 2. Let µk be the reflection at a sink/source k, then we have

isomorphism of invariant rings SIσ

α(Q) ∼

= SIµk(σ)

µk(α)(µk(Q)).

(1) So we obtain many equalities among Kronecker coefficients. (Question: how to find these equalities conceptually?) Moreover, the tensor invariants of U × V × W and U × V ′ × W are the same if dim U × dim W − dim V = dim V ′.

Jiarui Fei Tensor Invariants and Kronecker Coefficients

The Degree Bounds Conjecture

Conjecture

The degree bounds of the graded ring

• c∈N

SIcσ

α (K p,q m )U cλ is

polynomial in max( α, m) (or equivalently polynomial in max(l(λ), l(µ), l(ν))). This conjecture is equivalent to the relaxed modular index hypothesis in GCT VI, which implies the relaxed saturation

• hypothesis. So this conjecture is almost equivalent to that

we can decide whether kcλ

cµ,cν > 0 in poly(|λ|, |µ|, |ν|, c).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

The Degree Bounds Conjecture

Conjecture

The degree bounds of the graded ring

• c∈N

SIcσ

α (K p,q m )U cλ is

polynomial in max( α, m) (or equivalently polynomial in max(l(λ), l(µ), l(ν))). This conjecture is equivalent to the relaxed modular index hypothesis in GCT VI, which implies the relaxed saturation

• hypothesis. So this conjecture is almost equivalent to that

we can decide whether kcλ

cµ,cν > 0 in poly(|λ|, |µ|, |ν|, c).

Jiarui Fei Tensor Invariants and Kronecker Coefficients

Thank you!

Time for questions and comments

• Jiarui Fei

Tensor Invariants and Kronecker Coefficients