Complete finite semidirect products and wreath products

A complete group is one with a trivial center and with all automorphisms inner. This paper uses group cohomology to give a sufficient condition for a finite semidirect product G = N \rtimes H{G = N \rtimes H} with C _G (N) ≤ N to be complete and proves a partial converse. These results are enough to fully characterize complete finite permutational wreath products and to specialize that characterization in the case of finite standard wreath products.     

Representation theory of wreath products of finite groups

This is an introduction to the representation theory of wreath products of finite groups. We also discuss in full details a couple of examples. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Trees, wreath products and finite Gelfand pairs

We 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.     

Double centralizing theorem for wreath products of the alternating group

The double centralizing theorem between the action of the symmetric group S_n and the action of the general linear group on the tensor space T_n(W) was obtained by Schur. Here we obtain a double centralizing theorem when S_n is replaced by the wreath product of a finite group G and the alternating group A_n.     

On the embedding of central extensions into permutation wreath products

We find a necessary condition for embedding a central extension of a group G with elementary abelian kernel into the wreath product that corresponds to a given permutation action of G.     

On the representation theory of wreath products of finite groups and symmetric groups

Let G S_N be the wreath product of a finite group G and the symmetric group S_N. The aim of this paper is to prove the branching theorem for the increasing sequence of finite groups G S₁ G S₂ ... G S_N ... and the analog of Young's orthogonal form for this case, using the inductive approach invented by A. Vershik and A. Okounkov for the case of symmetric group.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 229–244.     

Zonal polynomials for wreath products

The pair of groups, symmetric group S _2n and hyperoctohedral group H _n , form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of (S _2n ,H _n ) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials. A wreath product generalization of the Gelfand pair (S _2n ,H _n ) is discussed in this paper. Then a multi-partition versions of the theory is constructed. The multi-partition version of zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the graded Hecke algebra into a multipartition version of the ring of symmetric functions. Dedicated to Professor Eiichi Bannai on his 60th birthday.