Rectangular Kronecker coefficients and plethysms in geometric complexity theory

TL;DR

This paper investigates the properties of rectangular Kronecker coefficients within geometric complexity theory, showing limitations for their use in proving lower bounds and establishing new positivity and classification results.

Contribution

It demonstrates the limitations of rectangular Kronecker coefficients in complexity lower bounds, proves their positivity in certain cases, and classifies their positivity behavior.

Findings

Rectangular Kronecker coefficients cannot prove superpolynomial lower bounds for the permanent.

Positivity of these coefficients is established for large partitions with quadratic side lengths.

Complete classification of positivity for rectangular limit Kronecker coefficients is provided.

Abstract

We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial. Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.