# Skew Howe duality and random rectangular Young tableaux

## TL;DR

This paper explores the decomposition of tensor products into irreducible components using skew Howe duality, revealing a probabilistic connection to random Young tableaux and analyzing their asymptotic behavior as dimensions grow large.

## Contribution

It establishes a novel probabilistic interpretation of the irreducible components in tensor decompositions via random Young tableaux, extending understanding to asymptotic regimes.

## Key findings

- Young diagram distribution matches entries ≤ p in a random rectangular Young tableau
- Asymptotic behavior of decomposition analyzed as m, n, p tend to infinity
- Skew Howe duality links tensor decomposition with random tableau models

## Abstract

We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(\lambda,\mu)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$\mu=\lambda'$. We show that the Young diagram $\lambda$ which corresponds to a randomly selected irreducible component $(\lambda,\lambda')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular shape with $m$ rows and $n$ columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as $m,n,p\to\infty$ tend to infinity.