Extended Bressoud–Wei and Koike skew Schur function identities

Abstract

The Jacobi-Trudi identity expresses a skew Schur function as a determinant of complete symmetric functions. Bressoud and Wei extend this idea, introducing an integer parameter t −1 and showing that signed sums of skew Schur functions of a certain shape are expressible once again as a determinant of complete symmetric functions. Koike provides a Jacobi-Trudi-style definition of universal rational characters of the general linear group and gives their expansion as a signed sum of products of Schur functions in two distinct sets of variables. Here we extend Bressoud and Wei's formula by including an additional parameter and extending the result to the case of all integer t. Then we introduce this parameter idea to the Koike formula, extending it in the same way. We prove our results algebraically using Laplace determinantal expansions.

The Pelletier-Ressayre hidden symmetry for Littlewood-Richardson coefficients

“…As mentioned above, Proposition 4.29 has the following generalization, which can be obtained from an identity of Koike [Koike89, Proposition 2.8] (via the map π n from [Koike89] and the correspondence between Schur Laurent polynomials and rational representations of GL (n)), which has later been extended by Hamel and King [HamKin11] (see [HamKin11,(6) and (10)] for the connection): 13…”

Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters

“…Proof. We follow a procedure introduced in [12], applied this time to a determinant whose entries involve a signed sum of a pair of elementary symmetric functions rather than as previously a weighted sum of a pair of complete homogeneous symmetric functions. The expansion of the determinant on the left of (A.19) yields where the sum is over those κ such that κ ℓ = 2ℓ for ℓ ∈ {ℓ 1 , ℓ 2 , .…”

Lattice Path Proofs of Extended Bressoud-Wei and Koike Skew Schur Function Identities

“…(41) The sign changing involution, which may be identified in general from the left-most pair of intersecting paths, is provided in this case by the transposition (5,6). The (n, m)-tuple contributes mutually cancelling monomials associated with the two permutations…”

“…Our recent paper [5] provides proofs of certain generalizations of two classical determinantal identities, one by Bressoud and Wei [1] and one by Koike [8]. Both of these identities are extensions of the Jacobi-Trudi identity, an identity that provides a determinantal representation of the Schur function.…”

“…We give the barest of background details and notation, referring the reader instead to our earlier paper [5], and to Macdonald [10] or Stanley [11] for general symmetric function background knowledge.…”