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A note on the Kaloujnine-Krasner theorem
No full textThe celebrated Kaloujnine-Krasner theorem associates, with a short exact sequence 1 -> N -> (iota) G -> (pi) H -> 1 of groups and a section s:H -> G, an embedding Phi : G -> N(sic)H of G into the (unrestricted) wreath product of N and H. Given two groups H and N, a short exact sequence as above is called an extension of H by N, denoted by (G;iota,pi). Moreover, one says that two extensions (G(1);iota(1),pi(1)) and (G(2);iota(2),pi(2)) of H by N are equivalent if there exists a group isomorphism eta : G(1) -> G(2) such that iota(2)=eta circle iota(1) and pi(1)=pi(2)circle eta. We say that two embeddings Phi(1):G(1) -> N(sic)H and Phi(2):G(2)-> N(sic)H are equivalent if there exists a group isomorphism eta : G(1) -> G(2) such that Phi(1)=Phi(2) circle eta. We show that two extensions (G(1);iota(1),pi(1)) and (G(2);iota(2),pi(2)) are equivalent if and only if the embeddings Phi(1) and Phi(2), associated with any two sections s(1 ): H -> G(1) and s(2 ): H -> G(2) via the Kaloujnine-Krasner theorem, are equivalent
Gelfand triples and their hecke algebras: harmonic analysis for multiplicity-free induced representations of finite groups
No full textTriple di Gelfand e loro algebre di HeckeGelfand triple and their Hecke algebra
Abelian extensions
No full textThis chapter is based on (Canad J Math 23:857–865, 1971; Canad J Math 25:1113–1119, 1973) by R. L. Roth. Previous papers with some results on this subject include (J. Fac. Sci. Univ. Tokyo Sec. I 10:129–146, 1964) (Pacific J Math 32:119–129, 1970) by N. Iwahori – H. Matsumoto and G. J. Janusz, respectively. See also Sect. 2.4, where the more general case when G∕IG(σ) is Abelian was studied
Trees, wreath products and finite Gelfand pairs
No full textWe present a new construction of finite Gelfand pairs by looking at the action of the full automorphism group of a finite spherically homogeneous rooted tree of type r on the variety V(r,s) of all spherically homogeneous subtrees of type s.
This generalizes well-known examples as the finite ultrametric space, the Hamming scheme and the Johnson scheme.
We also present further generalizations of these classical examples. The first two are based on Harary's notions of composition and exponentiation of group actions. Finally, the generalized Johnson scheme provides the inductive step for the harmonic analysis of our main construction
Projective representations of finite abelian groups with applications
No full textIn this chapter, we describe the irreducible projective representations of a finite Abelian group: we shall use the Clifford theory from the previous chapter. Then, by means of the machinery developed in Sect. 7.3, we describe the ordinary, irreducible representations of finite metabelian groups. As a particular case, we obtain an alternative description of the irreducible representations of finite 2-step nilpotent groups (cf. Sect. 6.6 )
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