Wreath products by the symmetric groups and product posets of Young’s lattices. (English) Zbl 0707.05062

Okada, Soichi

J. Comb. Theory, Ser. A 55, No. 1, 14-32 (1990).

The Young lattice is the poset of all partitions of the set of positive integers, namely almost everywhere null never increasing sequences of non-negative integers with positive sum. Studying the connections between wreath products \(G\wr S_ n\) of a finite group G with a symmetric groups \(S_ n\) and powers of the Young lattice, the author is able to give a complete set of mututally orthogonal eigenvectors for the linear mapping \(Ind^ n_{n-1}\circ Res^ n_{n-1}\) of the vector space of class functions of \(G\wr S_ n\), where \(Ind^ n_{n-1}\) is the induction mapping from (the vector space of class functions of) \(G\wr S_{n-1}\) to \(G\wr S_ n\) and \(Res^ n_{n-1}\) is the restriction mapping from (the vector space of class functions of) \(G\wr S_ n\) to \(G\wr S_{n-1}\).

Reviewer: A.Pasini

MSC:

05E25	Group actions on posets, etc. (MSC2000)
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures
06A07	Combinatorics of partially ordered sets
20C30	Representations of finite symmetric groups

Keywords:

partitions; symmetric groups; wreath products; Young lattice

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