Fundamental invariants of tensors, Latin hypercubes, and rectangular Kronecker coefficients

TL;DR

This paper investigates polynomial invariants of tensors, linking fundamental invariants, Latin hypercubes, and Kronecker coefficients, and demonstrates how a conjecture on Latin cubes influences positivity results in representation theory.

Contribution

It establishes a connection between Latin hypercube conjectures and the positivity of Kronecker coefficients, providing new insights into fundamental tensor invariants.

Findings

Proves a 3D analogue of the Alon--Tarsi conjecture implies positivity of Kronecker coefficients.

Links Latin hypercube conjecture to degree sequences of fundamental invariants.

Provides explicit values for degrees of fundamental invariants.

Abstract

We study polynomial SL-invariants of tensors, mainly focusing on fundamental invariants which are of smallest degrees. In particular, we prove that certain 3-dimensional analogue of the Alon--Tarsi conjecture on Latin cubes considered previously by B\"urgisser and Ikenmeyer, implies positivity of (generalized) Kronecker coefficients at rectangular partitions and as a result provides values for degree sequences of fundamental invariants.