On the codimension growth of G-graded algebras

TL;DR

This paper proves a conjectured inequality relating the codimension growth of a G-graded algebra to its identity component, showing that the overall growth is bounded by the square of the group order times the identity component's growth.

Contribution

It establishes a new upper bound for the codimension growth of G-graded algebras, confirming a conjecture by Bahturin and Zaicev.

Findings

Proved that exp(W)  |G|^2 exp(W_e)

Confirmed the conjecture by Bahturin and Zaicev

Provides a bound on codimension growth for G-graded PI-algebras

Abstract

Let W be an associative PI-affine algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. We prove: exp(W) \leq |G|^2 exp(W_e). This inequality had been conjectured by Bahturin and Zaicev.