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

Arvind Ayyer, Roger E. Behrend

Published 2018-04-12Version 1

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. Currently, we do not have representation-theoretic interpretations of any of the factorizations. Our results generalize the factorization identities obtained 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.