Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group

TL;DR

This paper establishes equivalences between the Alon-Tarsi conjecture, a Hadamard-Howe conjecture, and certain integrals over the special unitary group, motivated by geometric complexity theory.

Contribution

It demonstrates the equivalence of these conjectures and integrals, linking combinatorics, algebra, and geometry in a novel way.

Findings

Alon-Tarsi conjecture is equivalent to a Hadamard-Howe conjecture case.

Non-vanishing of specific SU(n) integrals relates to these conjectures.

Connections provide new insights into geometric complexity theory.

Abstract

We show the Alon-Tarsi conjecture on Latin squares is equivalent to a very special case of a conjecture made independently by Hadamard and Howe, and to the non-vanishing of some interesting integrals over SU(n). Our investigations were motivated by geometric complexity theory.