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Classical method of constructing a complete set of irreducible representations of semidirect product of a compact group with a finite group

Hirai T.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

353--362

Let G = U x S be a group of semidirect product of U compact and S finite. For an irreducible representation (= IR) ρ of U, let S([ρ]) be the stationary subgroup in S of the equivalence class [ρ] ∈ Û. Intertwining operators Jρ(s) (s ∈ S([ρ])) between ρ and s−1ρ gives in general a spin (= projective) representation of S([ρ]), which is lifted up to a linear representation J′ρ of a covering group S([ρ])′ of S([ρ]). Put π0:= ρ ·J′ρ, and take a spin representation π1 of S([ρ]) corresponding to the factor set inverse to that of Jρ, and put Π(π0, π1) = IndGU x S([ρ]) (π0 ▪ π1). We give a simple proof that Π(π0, π1) is irreducible and that any IR of G is equivalent to some of Π(π0, π1).

Indecomposable projective representations of direct products of finite groups over a ring of formal power series

Barannyk L. F.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2010

Vol. 15

9--24

Let F be a field of characteristic p > O, S = F[[X]] the ring of formal power series in the indeterminate X with coefficients in the field F, F* the multiplicative group of F, G = Gp x B a finite group, where Gp is a p-group and B is a p'-group. We give necessary and sufficient conditions for G and F under which there exists a cocycle λ ∈ Z2 (G, F*) such that every indecomposable projective 5-representation of G with the cocycle λ is the outer tensor product of an indecomposable projective 5-representation of Gp and an irreducible projective 5-representation of B.