On the realization of the Gelfand character of a finite group as a twisted trace
| Autor: | Jorge Soto-Andrade, Maria-Francisca Yáñez-Valdés |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Journal of Group Theory. 25:509-525 |
| ISSN: | 1435-4446 1433-5883 |
| Popis: | We show that the Gelfand character χ G \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 ↦ Tr ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ( G ) L^{2}(G) , where ( L 2 ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ( L ) \operatorname{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 ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions. |
| Databáze: | OpenAIRE |
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