Generic identities for finite group actions

TL;DR

This paper establishes a universal identity for finite group actions on free algebras, leading to optimal bounds in a longstanding problem about nilpotence in non-commutative rings, using graph theory techniques.

Contribution

It introduces a universal identity involving traces for finite group actions, providing the best nilpotence bounds in Bergman-Isaacs theorem through graph eigenvalue estimates.

Findings

Existence of a universal integer (G) for identities involving group actions.

Derivation of optimal nilpotence bounds in non-commutative ring actions.

Application of Cayley graph eigenvalue bounds to algebraic problems.

Abstract

Let $G$ be a finite group of order $n$, and $Z_G=\mathbb{Z}\langle\zeta_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the free generic algebra, with canonical action of $G$ according to $(\zeta_{i,g})^x=\zeta_{i,x^{-1}g}$. It is proved that there exists a positive integer $\upsilon(G)$ such that for any $g_1,g_2,\dots, g_{n}\in G$ $$ \upsilon(G)\cdot \zeta_{1,g_1}\zeta_{2,g_2}\dots\zeta_{n,g_{n}}=\sum_{i=1}^N \gamma_i a_i\mathbf{tr}_G(b_i)c_i, $$ where $\gamma_1,\gamma_2,\dots,\gamma_N$ are integers, and $a_i, b_i, c_i$ are monomials in $\zeta_{i,g}$ such that ${\rm deg}(b_i)>0$ and ${\rm deg}(a_i)+{\rm deg}(b_i)+{\rm deg}(c_i)=n$. As a consequence, if $R$ is a ring (not necessarily unital) acted on by $G$, then the product $\upsilon(G)\cdot R^{n}$ is contained in the ideal $\langle\mathbf{tr}_G(R)\rangle$ generated by all traces $\mathbf{tr}_G(r)=\sum\limits_{g\in G}r^g$, $r\in R$. This gives the best possible nilpotence bound in Bergman-Isaacs theorem for finite group actions on non-commutative rings, which was a long standing problem. The main result was obtained by transferring the problem to certain family of Cayley graphs, and estimating their minimal eigenvalues by the clique numbers. It is proved that the clique number $\omega(\Gamma)$ of any $k$-regular graph $\Gamma$ admits the Delsarte upper bound $\omega(\Gamma)\leqslant\lfloor1-k/\lambda_{\rm min}\rfloor$.