Twisted Frobenius–Schur indicators of finite symplectic groups

Vinroot, C. Ryan

doi:10.1016/j.jalgebra.2005.01.023

Cited by 11 publications

(17 citation statements)

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“…As an application of the results in the previous section, in this section we give a shorter and more direct proof of the main result of [42] regarding the finite symplectic groups Sp(2n, F q ) with q odd. For any q, define Sp 2n to be the collection of invertible transformations on V = F 2n q which stabilize the symplectic form ·, · defined by v, w = ⊤ v −I n I n w.…”

Section: Symplectic Groupsmentioning

confidence: 84%

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Totally orthogonal finite simple groups

Vinroot

2019

Math. Z.

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We prove that if G is a finite simple group, then all irreducible complex representations of G may be realized over the real numbers if and only if every element of G may be written as a product of two involutions in G. This follows from our result that if q is a power of 2, then all irreducible complex representations of the orthogonal groups O ± (2n, Fq) may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our main result for a broad family of finite classical groups. 2010

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Section: Symplectic Groupsmentioning

confidence: 84%

“…The result [42,Theorem 1.3] states that when q is odd, every irreducible complex character χ of Sp(2n, F q ) satisfies ε ι (χ) = 1. We give a condensed proof of this result by applying the methods and results obtained in the previous sections of this paper.…”

Section: Symplectic Groupsmentioning

confidence: 99%

Totally orthogonal finite simple groups

Vinroot

2019

Math. Z.

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“…Moreover, in [40] the authors finds a new model of the general linear group over a finite field (this construction can also be obtained from a result of Bannai, Kawanaka and Song [4] but the methods in [40] are independent of and different from theirs). • Vinroot [75] considers the group G = Sp(2n, F q ) equipped with the involutive automorphism g → −I n 0 0 I n g −I n 0 0 I n .…”

Section: Examplesmentioning

confidence: 99%

Mackey’s theory of $${\tau}$$ τ -conjugate representations for finite groups

Ceccherini‐Silberstein

Scarabotti

Tolli

2014

Jpn. J. Math.

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The aim of the present paper is to expose two contributions of Mackey, together with a more recent result of Kawanaka and Matsuyama, generalized by Bump and Ginzburg, on the representation theory of a finite group equipped with an involutory anti-automorphism (e.g. the anti-automorphism g → g −1 ). Mackey's first contribution is a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs. Mackey's second contribution is a characterization of simply reducible groups (a notion introduced by Wigner). The other result is a twisted version of the Frobenius-Schur theorem, where "twisted" refers to the above-mentioned involutory anti-automorphism. 42

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“…As it turns out, when q ≡ 1(mod 4) all irreducible complex characters of Sp(2n, F q ) are real-valued, but this is not the case when q ≡ 3(mod 4). This author unified these cases by proving [25,Theorem 1.3] that for any odd q, there is a certain twisted Frobenius-Schur indicator which is 1 for all irreducible complex characters of Sp(2n, F q ).…”

Section: Introductionmentioning

confidence: 98%