Multialternating graded polynomials and growth of polynomial identities

TL;DR

This paper constructs multialternating graded polynomials for finite dimensional G-graded algebras and determines the exponential growth rate of their graded codimensions, extending previous results to non-abelian groups.

Contribution

It introduces a method to compute the growth of graded identities for non-abelian G-graded algebras, generalizing prior abelian group results.

Findings

Constructed multialternating graded polynomials of arbitrarily large degree.

Computed the exponential growth rate of graded codimensions as an integer.

Extended known results from abelian to non-abelian groups.

Abstract

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.