Skew Howe duality and q-Krawtchouk polynomial ensemble

TL;DR

This paper analyzes the decomposition of an exterior algebra as a $GL_n imes GL_k$ module, revealing a probability distribution described by q-Krawtchouk polynomials, and establishes limit shape and fluctuation results as parameters grow.

Contribution

It introduces a novel probabilistic interpretation of the decomposition using q-Krawtchouk polynomial ensembles and derives asymptotic limit shapes and fluctuation theorems.

Findings

Probability distribution described by q-Krawtchouk polynomial ensemble.

Derived limit shape for the distribution as parameters grow.

Proved central limit theorem for fluctuations in the asymptotic regime.

Abstract

We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by pairs of Young diagrams $(\lambda,\bar{\lambda}')$, where $\bar{\lambda}'$ is the complement conjugate diagram to $\lambda$ inside the $n\times k$ rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the $q$-dimension for the irreducible component over the $q$-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when $n,k$ tend to infinity and $q$ tends to one at comparable rates.