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Fusion Procedure for the Symmetric Group Fusion procedure for the symmetric group Alexander Molev University of Sydney GL07, July 2007 The actions of Sk and glN commute with each other. Schur–Weyl duality The symmetric group Sk acts naturally on the tensor product space N N N C ⊗ C ⊗ ::: ⊗ C ; k factors; by permuting the factors. On the other hand, CN carries the vector representation of the Lie algebra glN so that the tensor product space is a representation of glN . Schur–Weyl duality The symmetric group Sk acts naturally on the tensor product space N N N C ⊗ C ⊗ ::: ⊗ C ; k factors; by permuting the factors. On the other hand, CN carries the vector representation of the Lie algebra glN so that the tensor product space is a representation of glN . The actions of Sk and glN commute with each other. L(λ) is the irreducible representation of glN with the highest weight λ, fλ is the dimension of the irreducible representation of Sk associated with λ. Irreducible decomposition of the glN -module N ⊗k ∼ L (C ) = fλ L(λ); λ where λ runs over partitions λ = (λ1; : : : ; λN ), λ1 > ··· > λN > 0 such that λ1 + ··· + λN = k, fλ is the dimension of the irreducible representation of Sk associated with λ. Irreducible decomposition of the glN -module N ⊗k ∼ L (C ) = fλ L(λ); λ where λ runs over partitions λ = (λ1; : : : ; λN ), λ1 > ··· > λN > 0 such that λ1 + ··· + λN = k, L(λ) is the irreducible representation of glN with the highest weight λ, Irreducible decomposition of the glN -module N ⊗k ∼ L (C ) = fλ L(λ); λ where λ runs over partitions λ = (λ1; : : : ; λN ), λ1 > ··· > λN > 0 such that λ1 + ··· + λN = k, L(λ) is the irreducible representation of glN with the highest weight λ, fλ is the dimension of the irreducible representation of Sk associated with λ. Refined decomposition N ⊗k ∼ L L N ⊗k (C ) = ΦU (C ) ; λ`k sh(U)=λ N ⊗k where each subspace LU = ΦU (C ) is a glN -submodule isomorphic to L(λ). fλ equals the number of standard λ-tableaux U. Let λ = (5; 3; 1), λ ` 9. The following λ-tableau U is standard 1 3 4 6 8 2 5 7 9 fλ equals the number of standard λ-tableaux U. Let λ = (5; 3; 1), λ ` 9. The following λ-tableau U is standard 1 3 4 6 8 2 5 7 9 Refined decomposition N ⊗k ∼ L L N ⊗k (C ) = ΦU (C ) ; λ`k sh(U)=λ N ⊗k where each subspace LU = ΦU (C ) is a glN -submodule isomorphic to L(λ). Problem: Find an explicit formula for the element φU 2 C[Sk ] whose image in the representation of Sk coincides with ΦU . r If U = U is the row tableau of shape λ, then the subspace LU r coincides with the image of the Young symmetrizer, N ⊗k LU r = HU r AU r (C ) ; where HU r and AU r are the row symmetrizer and column anti-symmetrizer of U r . r If U = U is the row tableau of shape λ, then the subspace LU r coincides with the image of the Young symmetrizer, N ⊗k LU r = HU r AU r (C ) ; where HU r and AU r are the row symmetrizer and column anti-symmetrizer of U r . Problem: Find an explicit formula for the element φU 2 C[Sk ] whose image in the representation of Sk coincides with ΦU . The orthonormal Young basis fvU g of Vλ is parameterized by the set of standard λ-tableaux U. Young basis Given a partition λ of k denote the corresponding irreducible representation of Sk by Vλ. The vector space Vλ is equipped with an Sk -invariant inner product ( ; ). Young basis Given a partition λ of k denote the corresponding irreducible representation of Sk by Vλ. The vector space Vλ is equipped with an Sk -invariant inner product ( ; ). The orthonormal Young basis fvU g of Vλ is parameterized by the set of standard λ-tableaux U. For any i 2 f1;:::; k − 1g set si = (i; i + 1). We have p s · v = d v + 1 − d 2 v ; i U U si U −1 where d = (ci+1 − ci ) , ci = ci (U) is the content b − a of the cell (a; b) occupied by i in a standard λ-tableau U, and the tableau si U is obtained from U by swapping the entries i and i + 1. Identify C[Sk ] with the direct sum of matrix algebras by fλ e 0 = φ 0 ; UU k! UU where φUU 0 is the matrix element corresponding to the basis vectors vU and vU 0 of the representation Vλ, X −1 φUU 0 = (s · vU ; vU 0 ) · s 2 C[Sk ]: s2Sk The group algebra C[Sk ] is isomorphic to the direct sum of matrix algebras ∼ L C[Sk ] = Matfλ (C); λ`k where fλ = dim Vλ. The matrix units eUU 0 2 Matfλ (C) are parameterized by pairs of standard λ-tableaux U and U 0. The group algebra C[Sk ] is isomorphic to the direct sum of matrix algebras ∼ L C[Sk ] = Matfλ (C); λ`k where fλ = dim Vλ. The matrix units eUU 0 2 Matfλ (C) are parameterized by pairs of standard λ-tableaux U and U 0. Identify C[Sk ] with the direct sum of matrix algebras by fλ e 0 = φ 0 ; UU k! UU where φUU 0 is the matrix element corresponding to the basis vectors vU and vU 0 of the representation Vλ, X −1 φUU 0 = (s · vU ; vU 0 ) · s 2 C[Sk ]: s2Sk 2 Since e U e V = 0 for U 6= V, e U = e U , and X X 1 = e U ; λ`k sh(U)=λ the elements we want are φU (or e U ), yielding N ⊗k ∼ L L N ⊗k (C ) = ΦU (C ) : λ`k sh(U)=λ For the diagonal elements we write e U = e UU and φU = φUU : the elements we want are φU (or e U ), yielding N ⊗k ∼ L L N ⊗k (C ) = ΦU (C ) : λ`k sh(U)=λ For the diagonal elements we write e U = e UU and φU = φUU : 2 Since e U e V = 0 for U 6= V, e U = e U , and X X 1 = e U ; λ`k sh(U)=λ For the diagonal elements we write e U = e UU and φU = φUU : 2 Since e U e V = 0 for U 6= V, e U = e U , and X X 1 = e U ; λ`k sh(U)=λ the elements we want are φU (or e U ), yielding N ⊗k ∼ L L N ⊗k (C ) = ΦU (C ) : λ`k sh(U)=λ The vectors of the Young basis are eigenvectors for the action of xi on Vλ. For any standard λ-tableau U we have xi · vU = ci (U) vU ; i = 1;:::; k: The Jucys–Murphy elements of C[Sk ] are defined by x1 = 0; xi = (1 i) + (2 i) + ··· + (i − 1 i); i = 2;:::; k: They generate a commutative subalgebra of C[Sk ]. Moreover, xk commutes with all elements of Sk−1. The Jucys–Murphy elements of C[Sk ] are defined by x1 = 0; xi = (1 i) + (2 i) + ··· + (i − 1 i); i = 2;:::; k: They generate a commutative subalgebra of C[Sk ]. Moreover, xk commutes with all elements of Sk−1. The vectors of the Young basis are eigenvectors for the action of xi on Vλ. For any standard λ-tableau U we have xi · vU = ci (U) vU ; i = 1;:::; k: The branching properties of the Young basis imply the corresponding properties of the matrix units. If V is a given standard tableau with the entries 1;:::; k − 1 then X eV = e U ; V!U where V!U means that the standard tableau U is obtained from V by adding one cell with the entry k. and we have the identity in C[Sk ], X X xk = ck (U) e U ; λ`k sh(U)=λ so that xk can be viewed as a diagonal matrix. Furthermore, xi e U = e U xi = ci (U) e U ; i = 1;:::; k for any standard λ-tableau U, Furthermore, xi e U = e U xi = ci (U) e U ; i = 1;:::; k for any standard λ-tableau U, and we have the identity in C[Sk ], X X xk = ck (U) e U ; λ`k sh(U)=λ so that xk can be viewed as a diagonal matrix. Murphy’s formula. We have the relation in C[Sk ], (xk − a1) ::: (xk − al ) e U = eV ; (c − a1) ::: (c − al ) where a1;:::; al are the contents of all addable cells of µ except for α, while c is the content of the latter. Equivalently, u − c e U = eV : u − xk u=c Now let k > 2 and let λ be a partition of k. Fix a standard λ-tableau U and denote by V the standard tableau obtained from U by removing the cell α occupied by k. Denote the shape of V by µ. Equivalently, u − c e U = eV : u − xk u=c Now let k > 2 and let λ be a partition of k. Fix a standard λ-tableau U and denote by V the standard tableau obtained from U by removing the cell α occupied by k. Denote the shape of V by µ. Murphy’s formula. We have the relation in C[Sk ], (xk − a1) ::: (xk − al ) e U = eV ; (c − a1) ::: (c − al ) where a1;:::; al are the contents of all addable cells of µ except for α, while c is the content of the latter.
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