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

## TL;DR

This paper explores the combinatorial and probabilistic structures of classical group characters, introduces new interpolating polynomials, and connects these to Last Passage Percolation models and Tracy-Widom distributions.

## Contribution

It introduces new interpolating polynomials between classical group characters, develops Gelfand-Tsetlin pattern decompositions, and links these to LPP models and random matrix theory distributions.

## Key findings

- Identified two families of symmetric polynomials interpolating between classical characters.
- Established identities and branching rules connecting orthogonal, symplectic, and interpolating polynomials.
- Connected LPP models with symmetries to Tracy-Widom GOE and GSE distributions.

## Abstract

We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_n(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$ and between characters of ${\rm SO}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$. We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of irreducible characters, can be viewed as branching rules. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains (Duke Math. J., 2001). Taking the scaling limit of the LPP models, we finally 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.