Lattic path proofs of extended Bressoud-Wei and Koike skew Schur function identities

Our recent paper provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs

Extended Bressoud-Wei and Koike skew Schur function identities

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 $t\geq-1$ 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

Lattice path proofs of extended Bressoud-Wei and Koike skew Schur function identities

Our recent paper \cite{HK10} provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs

The Pelletier-Ressayre hidden symmetry for Littlewood-Richardson coefficients

We prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre (arXiv:2005.09877). The proof relies on a novel birational involution defined over any semifield.Comment: 85 pages. Reproves some basic properties of Schur Laurent polynomials (with negative parts) for apparent lack of explicit coverage in the literature. The birational map studied in Section 3 might be useful elsewhere. Detailed version available as ancillary file. v5 adds Subsection 5.4 on the birational R-matrix. Comments are welcome