Discrete Unitary Invariance

TL;DR

This paper demonstrates that specific determinantal functions of multiple matrices exhibit invariance under cube symmetries, decomposing into functions of the original matrices in a general algebraic setting.

Contribution

It establishes a broad invariance property of determinantal functions under symmetry groups without relying on vector space properties.

Findings

Determinantal functions are invariant under cube symmetries.

Invariance holds over any commutative ring.

Proofs are elementary and derivation-based.

Abstract

We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of the underlying vector space will be used apart from normal ring properties, and therefore hold in any commutative ring. All proofs are elementary --- in fact, the majority are simply derivations.