Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

TL;DR

This paper establishes new factorization theorems for classical group characters, specifically Schur polynomials, and applies these results to derive identities related to alternating sign matrices and plane partitions.

Contribution

It generalizes previous factorization identities for Schur polynomials and introduces new factorizations involving sums of two Schur polynomials and specific variable sets.

Findings

Factorization of Schur polynomials into orthogonal characters for certain partitions and variables

Derivation of identities for counts of plane partitions and alternating sign matrices

Extension of previous results to broader classes of partitions and variable configurations

Abstract

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (Advances in combinatorial mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangular or double-staircase shapes and all of the variables are set to 1, then factorization identities for numbers of certain plane partitions, alternating sign matrices and related combinatorial objects are obtained.