Tensor Invariants and Kronecker Coefficients Jiarui Fei University - PowerPoint PPT Presentation

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 ( n 1 , n 2 )-dimensional representation space of the m -arrow Kronecker quiver K m , where dim U = n 1 , dim V = n 2 , and dim W = m . The ring of invariants SI ( n 1 , n 2 ) ( K m ) := 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 ( n 1 , n 2 )-dimensional representation space of the m -arrow Kronecker quiver K m , where dim U = n 1 , dim V = n 2 , and dim W = m . The ring of invariants SI ( n 1 , n 2 ) ( K m ) := 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 n 1 = n 2 = n . By a semi-invariants of degree d , we mean an element f ∈ SI ( n , n ) ( K m ) of weight ( d , − d ), i.e., for any x ∈ U ∗ ⊗ V ⊗ W ∗ and ( g 1 , g 2 ) ∈ GL( U ) × GL( V ) we have that f (( g 1 , g 2 ) x ) = det( g 1 ) d det( g 2 ) − d f ( x ) . We consider the determinant of the dn × dn matrix m � Λ k ⊗ A k , k =1 where Λ k = ( λ k ij ) is a d × d matrix and ⊗ is the Kronecker product. ij ) ( D k ) ij in the i , j , k ( λ k Let c ( D 1 , D 2 , . . . , D m ) be the coefficient of � expansion. Jiarui Fei Tensor Invariants and Kronecker Coefficients

Variation of Schofield’s Invariants For simplicity, we will assume n 1 = n 2 = n . By a semi-invariants of degree d , we mean an element f ∈ SI ( n , n ) ( K m ) of weight ( d , − d ), i.e., for any x ∈ U ∗ ⊗ V ⊗ W ∗ and ( g 1 , g 2 ) ∈ GL( U ) × GL( V ) we have that f (( g 1 , g 2 ) x ) = det( g 1 ) d det( g 2 ) − d f ( x ) . We consider the determinant of the dn × dn matrix m � Λ k ⊗ A k , k =1 where Λ k = ( λ k ij ) is a d × d matrix and ⊗ is the Kronecker product. ij ) ( D k ) ij in the i , j , k ( λ k Let c ( D 1 , D 2 , . . . , D m ) be the coefficient of � expansion. Jiarui Fei Tensor Invariants and Kronecker Coefficients

Example ( m = 2 , d = 3) λ 1 11 A 1 + λ 2 λ 1 12 A 1 + λ 2 λ 1 13 A 1 + λ 2   11 A 2 12 A 2 13 A 2 λ 1 21 A 1 + λ 2 λ 1 22 A 1 + λ 2 λ 1 23 A 1 + λ 2 21 A 2 22 A 2 23 A 2   λ 1 31 A 1 + λ 2 λ 1 32 A 1 + λ 2 λ 1 33 A 1 + λ 2 31 A 2 32 A 2 33 A 2 Lemma { c ( D 1 , D 2 , . . . , D m ) } linearly span the quiver invariants SI d ( n , n ) ( K m ) . Jiarui Fei Tensor Invariants and Kronecker Coefficients

Example ( m = 2 , d = 3) λ 1 11 A 1 + λ 2 λ 1 12 A 1 + λ 2 λ 1 13 A 1 + λ 2   11 A 2 12 A 2 13 A 2 λ 1 21 A 1 + λ 2 λ 1 22 A 1 + λ 2 λ 1 23 A 1 + λ 2 21 A 2 22 A 2 23 A 2   λ 1 31 A 1 + λ 2 λ 1 32 A 1 + λ 2 λ 1 33 A 1 + λ 2 31 A 2 32 A 2 33 A 2 Lemma { c ( D 1 , D 2 , . . . , D m ) } linearly span the quiver invariants SI d ( n , n ) ( K m ) . Jiarui Fei Tensor Invariants and Kronecker Coefficients

Relations We view c ( D 1 , D 2 , . . . , D m ) as formal variables, and denote � � ( λ k ij ) ( D k ) ij . F ( C , Λ) := c ( D 1 , D 2 , . . . , D m ) i , j , k Since we have for any elementary matrix F ij = I + E ij that, � � det( F ij Λ k ⊗ A k ) = det( Λ k ⊗ A k ) , k k such an equality generates a set of relations among c ( D 1 , D 2 , . . . , D m ), namely F ( C , F ij Λ) − F ( C , Λ) = 0 . Theorem The relations below span all relations among c ( D 1 , D 2 , . . . , D m ) ’s. F ( C , F ij Λ) − F ( C , Λ) = 0 , and F ( C , Λ F ij ) − F ( C , Λ) = 0 . Jiarui Fei Tensor Invariants and Kronecker Coefficients

Relations We view c ( D 1 , D 2 , . . . , D m ) as formal variables, and denote � � ( λ k ij ) ( D k ) ij . F ( C , Λ) := c ( D 1 , D 2 , . . . , D m ) i , j , k Since we have for any elementary matrix F ij = I + E ij that, � � det( F ij Λ k ⊗ A k ) = det( Λ k ⊗ A k ) , k k such an equality generates a set of relations among c ( D 1 , D 2 , . . . , D m ), namely F ( C , F ij Λ) − F ( C , Λ) = 0 . Theorem The relations below span all relations among c ( D 1 , D 2 , . . . , D m ) ’s. F ( C , F ij Λ) − F ( C , Λ) = 0 , and F ( C , Λ F ij ) − 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 E kl ∈ sl ( W ). Then we have the following formula � � � � � ∂ kl c ( D 1 , . . . , D m ) = − ( D l ) ij +1 c ( . . . , D k − E ij , . . . , D l + E ij , . . . ) ( D k ) ij > 0 In particular, c ( D ) is T -invariant iff. n n D ij = dn � � m | dn and m (1 , 1 , . . . , 1) . i =1 j =1 n m n m � � � � (In priori, D k = D k = n (1 , 1 , . . . , 1) . ) i =1 k =1 j =1 k =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 E kl ∈ sl ( W ). Then we have the following formula � � � � � ∂ kl c ( D 1 , . . . , D m ) = − ( D l ) ij +1 c ( . . . , D k − E ij , . . . , D l + E ij , . . . ) ( D k ) ij > 0 In particular, c ( D ) is T -invariant iff. n n D ij = dn � � m | dn and m (1 , 1 , . . . , 1) . i =1 j =1 n m n m � � � � (In priori, D k = D k = n (1 , 1 , . . . , 1) . ) i =1 k =1 j =1 k =1 Jiarui Fei Tensor Invariants and Kronecker Coefficients

Finite-type Cases 222 The degree 2 invariant is Cayley’s hyperdeterminant c (11) 2 − 4 c (20) c (02) . 233 The degree 4 invariant is c (12) 2 c (21) 2 − 4 c (03) c (21) 3 − 4 c (12) 3 c (30) +18 c (03) c (12) c (21) c (30) − 27 c (03) 2 c (30) 2 . 223 The degree 3 invariant is � 000 001 100 � 000 001 100 � � 0 A 3 − A 2 � � c − c = det . − A 3 0 A 1 001 000 010 100 000 010 010 100 000 010 001 000 − A 1 0 A 2 224 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) ( K 4 ) is free over Sym( S 2 ) with basis S (1 4 ) , and has equivariant free resolution S (2 4 ) ⊗ 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 � c ( 020 100 100 002 ) + c ( 002 010 010 200 ) + c ( 200 001 � 000 111 ) + 4 001 020 ) . � � 0 A 3 − A 2 The degree 3 invariant is det . − A 3 0 A 1 − A 1 0 A 2 The degree 4 invariant is the Aronhold invariant of ternary cubic form. The invariant ring SI (3 , 3) ( K 3 ) is free over Sym( S 3 ⊕ S (2 3 ) ) with basis S (3 3 ) , and has equivariant free resolution S (6 3 ) ⊗ A ( − 6) → A , where A = Sym( S 3 ⊕ S (2 3 ) ⊕ S (3 3 ) ) . 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 � c ( 020 100 100 002 ) + c ( 002 010 010 200 ) + c ( 200 001 � 000 111 ) + 4 001 020 ) . � � 0 A 3 − A 2 The degree 3 invariant is det . − A 3 0 A 1 − A 1 0 A 2 The degree 4 invariant is the Aronhold invariant of ternary cubic form. The invariant ring SI (3 , 3) ( K 3 ) is free over Sym( S 3 ⊕ S (2 3 ) ) with basis S (3 3 ) , and has equivariant free resolution S (6 3 ) ⊗ A ( − 6) → A , where A = Sym( S 3 ⊕ S (2 3 ) ⊕ S (3 3 ) ) . 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 � k λ S µ S ν = µ,ν S λ . λ By Schur-Weyl duality, it also appears as S λ ( V ⊗ W ) ∼ � k λ µ,ν S µ ( V ) ⊗ S ν ( W ) . = µ,ν Jiarui Fei Tensor Invariants and Kronecker Coefficients