Spin representations of twisted central products of double covering finite groups and the case of permutation groups

Abstract

Let S be a finite group with a character, sgn, of order 2, and S′ its central extension by a group Z = ⟨z⟩ of order 2. A representation π of S′ is called spin if π(zσ′) = −π(σ′) (σ′ ∈ S′), and the set of all equivalence classes of spin irreducible representations (= IRs) of S′ is called the spin dual of S′. Take a finite number of such triplets (S_j′, z_j, sgn_j) (1 ≤ j ≤ m). We define twisted central product S′ = S₁′ $\hat{*}$ S₂′ $\hat{*}$ … $\hat{*}$ S_m′ as a double covering of S = S₁ × … × S_m, S_j = S_j′/⟨z_j⟩, and for spin IRs π_j of S_j′, define twisted central product π = π₁ $\hat{*}$ π₂ $\hat{*}$ … $\hat{*}$ π_m as a spin IR of S′. We study their characters and prove that the set of spin IRs π of this type gives a complete set of representatives of the spin dual of S′. These results are applied to the case of representation groups S′ for S = 𝔖_n and 𝔄_n, and their (Frobenius-)Young type subgroups.