Nonvanishing of Kronecker coefficients for rectangular shapes

TL;DR

This paper proves the nonvanishing of certain Kronecker coefficients for rectangular shapes, demonstrating their existence under specific conditions, which advances understanding in geometric complexity theory.

Contribution

It establishes the existence of nonzero Kronecker coefficients for rectangular partitions and provides bounds on the stretching factor, contributing to geometric complexity theory.

Findings

Existence of nonvanishing Kronecker coefficients for rectangular shapes

Bounded stretching factor in approximate nonvanishing results

Implications for geometric complexity theory

Abstract

We prove that for any partition $(\lambda_1,...,\lambda_{d^2})$ of size $\ell d$ there exists $k\ge 1$ such that the tensor square of the irreducible representation of the symmetric group $S_{k\ell d}$ with respect to the rectangular partition $(k\ell,...,k\ell)$ contains the irreducible representation corresponding to the stretched partition $(k\lambda_1,...,k\lambda_{d^2})$. We also prove a related approximate version of this statement in which the stretching factor $k$ is effectively bounded in terms of $d$. This investigation is motivated by questions of geometric complexity theory.