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• W3209658051 abstract "Abstract We show that the Gelfand character <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>χ</m:mi> <m:mi>G</m:mi> </m:msub> </m:math> 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” <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>g</m:mi> <m:mo>↦</m:mo> <m:mrow> <m:mi>Tr</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msub> <m:mi>ρ</m:mi> <m:mi>g</m:mi> </m:msub> <m:mo>∘</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> gmapstooperatorname{Tr}(rho_{g}circ T) for a suitable involutive linear automorphism 𝑇 of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> L^{2}(G) , where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>ρ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> (L^{2}(G),rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>T</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>f</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>∘</m:mo> <m:mi>L</m:mi> </m:mrow> </m:mrow> </m:math> T(f)=fcirc L ( <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> fin 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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>Fix</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>L</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msub> <m:mi>Fix</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>L</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>τ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:math> (L^{2}(operatorname{Fix}_{G}(L)),tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions." @default.
• W3209658051 created "2021-11-08" @default.
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• W3209658051 date "2021-10-28" @default.
• W3209658051 modified "2023-10-16" @default.
• W3209658051 title "On the realization of the Gelfand character of a finite group as a twisted trace" @default.
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• W3209658051 doi "https://doi.org/10.1515/jgth-2020-0207" @default.
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