Abstract
The multiplicity-free subgroups (strong Gelfand subgroups) of wreath products are investigated. Various useful reduction arguments are presented. In particular, we show that for every finite group [Formula: see text], the wreath product [Formula: see text], where [Formula: see text] is a Young subgroup, is multiplicity-free if and only if [Formula: see text] is a partition with at most two parts, the second part being 0, 1, or 2. Furthermore, we classify all multiplicity-free subgroups of hyperoctahedral groups. Along the way, we derive various decomposition formulas for the induced representations from some special subgroups of hyperoctahedral groups.
Highlights
Spinn−1(C)) is a multiplicity-free subgroup in SLn(C) (resp. in Spinn(C)). (SLn(C), GLn−1(C)) and (Spinn, Spinn−1) are strong Gelfand pairs. It was shown by Kramer in [15] that for simple, connected algebraic groups, there are no additional pairs of strong Gelfand pairs
We summarize the conclusions of the previous subsections in a single proposition
Let K be a subgroup of Bn such that γK = Sn−1 × S1
Summary
The strongest results of our paper are about the pairs with F = Z/2, we prove some general theorems when F is a finite (abelian) group. The first novel results of our paper appear, where 1) we prove a key lemma that we use later for describing some branching rules in wreath products, 2) we describe the multiplicities of the irreducible representations in indFSnSn Sλ where F is an abelian group, and Sλ is a Specht module of Sn labeled by the partition λ of n. By assuming that F is an abelian group, we prove a stronger statement in one direction: (F Sn, (F Sn−k) × Sk) is a strong Gelfand pair if k 2. The strong Gelfand subgroups of B2 are given in Lemma 7.48; those of B3 are given in Proposition 7.74, in which we give the number of strong Gelfand subgroups of Bn for each 4 n 7
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
