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

DOI10.1515/JGTH-2020-0207arXiv1303.4800MaRDI QIDQ6240469

Maria-Francisca Yáñez-Valdés, Jorge Soto Andrade

Publication date: 19 March 2013

Abstract: We show that the Gelfand character

c h i_{G}

of a finite group

G

(i.e. the sum of all irreducible complex characters of

G

) may be realized as a `` twisted trace $g m a p s t o T r (h o_{g} c i r c T)$ for a suitable involutive linear automorphism $T$ of $L^{2} (G)$ , where $(L^{2} (G), h o)$ is the right regular representation of $G$ . Moreover, we prove that under certain hypotheses $T (f) = f c i r c L;; (f i n L^{2} (G)),$ where $L$ is an involutive antiautomorphism of $G .$ The natural representation $a u$ of $G$ associated to the natural $L$ -conjugacy action of $G$ in the fixed point set $F i x_{G} (L)$ of $L$ turns out to be a Gelfand Model for $G$ in some cases. We show that $(L^{2} (F i x_{G} (L)), a u)$ fails to be a Gelfand Model if $G$ admits non trivial central involutions.

Mathematics Subject Classification ID

Ordinary representations and characters (20C15)

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