Representations of the Hyperoctahedral Group B _n

Abstract

In this chapter, we shall repeat the outlay of Chapter 5 for the Hyperoctahedral subgroup B _n (7.1.0) of S_2n under similar assumptions as for S _n . That is to determine all irreducible representations of B _n over an algebraically closed field K such that K[B _n ] is semi-simple. We shall see that B _n is of order 2ⁿn! (7.1.1) and hence K[B _n ] is semi-simple if the characteristic of K is either 0 or a prime > n (in which case K[S_2n] may not be semi-simple). As remarked in (3.7.6) above and carried out for the symmetric group S _n in Chapter 5, we need to do four things, namely, (i) determine the conjugacy classes of B _n , (ii) for each conjugacy class (λ,μ), construct an irreducible representation V_(λ,μ) in such a way that (iii) V_(λ,μ) is not equivalent to V_(λ,δ) for (λ,μ) ≠ (γ,δ) and (iv) determine the dimensions of the V_(λ,μ)’s.