Polynomial identities for matrices over the Grassmann algebra

TL;DR

This paper establishes minimal Cayley-Hamilton and Capelli identities for matrices over finite-rank Grassmann algebras, providing bounds that improve upon previous results in the literature.

Contribution

It determines minimal identities for matrices over Grassmann algebras and refines bounds on their degrees, advancing understanding of polynomial identities in this context.

Findings

Derived minimal Cayley-Hamilton identities for Grassmann algebra matrices

Established bounds on degrees of minimal standard identities

Improved previous upper bounds on polynomial identities

Abstract

We determine minimal Cayley--Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a recent paper.