Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

Elia Bisi, Nikos Zygouras

Published 2019-05-23Version 1

We introduce two families of symmetric polynomials that interpolate between irreducible characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$ and between irreducible characters of ${\rm SO}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$. We define them as generating functions of certain kinds of Gelfand-Tsetlin patterns and establish `Weyl character formulas' as ratios of determinants, among other properties. Via a method of pattern decomposition, we establish identities between these interpolating polynomials and also between characters of the classical groups, which can be viewed as describing the decomposition of irreducible representations of the groups when restricted to certain subgroups. Through such formulas, we also connect the orthogonal and symplectic characters, as well as our interpolating polynomials, to the probabilistic model of last passage percolation with various symmetries; we thus go beyond the link with the general linear characters (i.e. classical Schur polynomials) originally found by Baik and Rains [BR01a]. As an application, we provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.