On Involutions and Indicators of Finite orthogonal Groups

Abstract

We study the numbers of involutions and their relation to Frobenius-Schur indicators in the groups SO ± (n, q) and Ω ± (n, q). Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group G is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius-Schur indicators are 1), and we are able to show this holds for all finite simple groups G other than the groups Sp(2n, q) with q even or Ω ± (4m, q) with q even. We prove computationally that for small n and m this statement indeed holds for these groups by equating their character degree sums to the number of involutions. We also prove a result on a certain twisted indicator for the groups SO ± (4m + 2, q) with q odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating function and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in SO ± (n, q) and Ω ± (n, q) for all q, and we use these to compute the asymptotic behavior for the number of involutions in these groups when q is fixed and n grows.2010 AMS Subject Classification: 20G40, 20C33, 05A15, 05A16

Real Representations of Finite Symplectic Groups over Fields of Characteristic Two

“…Also relevant to our main result in Theorem 4.2 is the following conjecture regarding finite simple groups, first stated explicitly in [15], and discussed in [24,Section 3]: A finite simple group G has the property that all of its complex irreducible representations are real representations, if and only if every element of G is the product of two involutions from G. Our Theorem 4.2 resolves one of the remaining families to check for this conjecture to hold. We hope to extend the methods below to confirm this conjecture for the final cases in a subsequent paper.…”

“…Meanwhile, Fulman, Guralnick, and Stanton have computed a generating function for the number of involutions in G = SO(2n + 1, F q ) (where G = 1 if n = 0). Specifically, it follows from [8, Theorem 2.17 and Lemma 6.1(3)] and the fact that O(2n + 1, F q ) ∼ = SO(2n + 1, F q ) × {±1} (see also [24,Theorem 7.1(2)]), that the number of…”

“…A result of Prasad [23,Theorem 3] implies that any complex irreducible representation of Sp(2n, F q ) with q even, which appears in the Gelfand-Graev representation, can be defined over the real numbers. It has also been shown computationally [24,Theorem 5.2] than when q is even and n ≤ 8, all complex irreducible representations of Sp(2n, F q ) are real representations. In this paper, we finally settle the general case, and we show in Theorem 4.2 that when q is even every complex irreducible representation of Sp(2n, F q ), for any n, is a real representation.…”

Totally orthogonal finite simple groups

“…In Section 2.2, we give background on the standard and twisted Frobenius-Schur indicator, including the crucial (twisted) involution formula which says that the character degree sum for a finite group is equal to the number of (twisted) involutions if and only if all (twisted) indicators are 1. In Section 3 we give the generating functions for the number of involutions in the finite orthogonal groups due to Fulman, Guralnick, and Stanton [13], and for the number of involutions in the finite special orthogonal groups or in its other coset in the orthogonal groups, from [37]. These generating functions motivate our main method in the following way.…”

“…Other than several alternating groups and two sporadic groups, the rest of the groups on the list of strongly real finite simple groups are groups of Lie type. When one considers whether these groups are totally orthogonal, as explained in [37,Section 3], this follows for many of the groups from previous work, like the paper of Gow [17] which covers the simple symplectic and orthogonal groups in odd characteristic. The only cases which do not follow from previous results are the symplectic groups Sp(2n, F q ) when q is a power of 2, which this author proved in [44], and the special orthogonal groups Ω ± (4m, F q ) = SO ± (4m, F q ) when q is a power of 2, which we complete here in Theorem 8.3.…”