A graph theoretic approach to graded identities for matrices

TL;DR

This paper introduces a graph-theoretic method to analyze graded identities in matrix algebras with crossed product gradings, providing new proofs and an asymptotic formula for graded codimensions.

Contribution

It develops a novel graph-based approach to study graded identities in matrix algebras and determines the asymptotic behavior of G-graded codimensions.

Findings

New graph-theoretic proofs of known identity generators.

Determination of the asymptotic formula for G-graded codimension.

Enhanced understanding of graded polynomial identities in matrix algebras.

Abstract

We consider the algebra M_k(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by embedding G in the symmetric group S_k via the regular representation and embedding S_k in M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in G. We study the graded polynomial identities for M_k(C) equipped with a crossed product grading. To each multilinear monomial in the free graded algebra we associate a directed labeled graph. This approach allows us to give new proofs of known results of Bahturin and Drensky on the generators of the T-ideal of identities and the Amitsur-Levitsky Theorem. Our most substantial new result is the determination of the asymptotic formula for the G-graded codimension of M_k(C).