On the realization of the Gelfand Character of a finite group as a twisted trace

TL;DR

This paper demonstrates how the Gelfand character of a finite group can be expressed as a twisted trace involving an involutive automorphism, providing new insights into Gelfand models and their limitations.

Contribution

It introduces a method to realize the Gelfand character as a twisted trace using an involutive automorphism and explores conditions under which a natural representation forms a Gelfand model.

Findings

Gelfand character can be expressed as a twisted trace involving an involutive automorphism.

The natural representation associated with the L-conjugacy action can serve as a Gelfand model in some cases.

The Gelfand model fails when the group has non-trivial central involutions.

Abstract

We show that the Gelfand character $ \chi_G$ of a finite group $G $ (i.e. the sum of all irreducible complex characters of $G$ ) may be realized as a `` twisted trace'' $ g \mapsto Tr( \rho_g \circ T) $ for a suitable involutive linear automorphism $T$ of $L^2(G)$, where $(L^2(G), \rho)$ is the right regular representation of $G$. Moreover, we prove that under certain hypotheses $T(f)= f \circ L \;\; (f \in L^2(G)), $ where $ L $ is an involutive antiautomorphism of $ G.$ The natural representation $\tau$ of $G$ associated to the natural $L$-conjugacy action of $G$ in the fixed point set $Fix_G(L)$ of $L$ turns out to be a Gelfand Model for $G$ in some cases. We show that $(L^2(Fix_G(L)), \tau) $ fails to be a Gelfand Model if $G$ admits non trivial central involutions.