# A $q$-deformation of the symplectic Schur functions and the Berele insertion algorithm

## TL;DR

This paper introduces a probabilistic $q$-deformation of Berele's insertion algorithm for symplectic Young tableaux, enabling new identities for symplectic Schur functions and analyzing their shape evolution as a Markov chain.

## Contribution

It develops a $q$-deformed insertion algorithm for symplectic tableaux, extending Berele's classical algorithm and connecting to $q$-Schur functions and Markov chain dynamics.

## Key findings

- Introduces a $q$-deformed Berele insertion algorithm.
- Proves Littlewood-type identities for $q$-deformed symplectic Schur functions.
- Shows the shape evolution as a Markov chain on partitions.

## Abstract

A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele's algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0 < q < 1$. The classic Berele algorithm corresponds to letting the parameter $q \to 0$. The new version provides a probabilistic framework that allows to prove Littlewood-type identities for a $q$-deformation of the symplectic Schur functions. These functions correspond to multilevel extensions of the continuous $q$-Hermite polynomials. Finally, we show that when both the original and the $q$-modified insertion algorithms are applied to a random word then the shape of the symplectic Young tableau evolves as a Markov chain on the set of partitions.