Generalized Frobenius–Schur numbers

Involution models of finite Coxeter groups

“…The group Alt(5) has an involution model, as given in [4,Sec. 3], and since Z/2Z has an involution model, it follows from Lemma 1 that the group of type H 3 has an involution model.…”

Automorphisms and generalized involution models of finite complex reflection groups

“…(b) The image of ι contains exactly one element from each τ -twisted conjugacy class in I G,τ . This definition differs slightly from the one originally given in [9], but one can show that it is equivalent, in the sense that the same models are classified as generalized involution models.…”

Twisted Frobenius–Schur indicators of finite symplectic groups

“…Then ε(π) is exactly the Frobenius-Schur indicator ε(χ) in Theorem 2.1. This is proven in a much more general setting by Bump and Ginzburg [1], but we now only state the generalization that we need, which is the case corresponding to Theorem 2.2 of Kawanaka and Matsuyama. Let ι be an order 2 automorphism of the finite group G, and let (π, V ) be a complex irreducible representation of G such that ι π ∼ =π , where ι π(g) = π( ι g).…”

“…Let µ be the similitude character, and ι the inner automorphism of GSp(2n, F q ) which conjugates by the skew-symplectic element as in (1). Define σ to be the order 2 automorphism of GSp(2n, F q ) which acts as σ g = µ(g) −1 · ι g. Then the main result for this group is the following, with the rather surprising result for the sum of the character degrees.…”

“…Theorem 1.3. Let q be odd, and ι the automorphism of Sp(2n, F q ) as in (1). Then every irreducible representation π of Sp(2n, F q ) satisfies ε ι (π) = 1.…”