Abstract :
Hecke algebras for the complex reflection groups G(r,p,n) (where p 1 and p r) were introduced in the work of Ariki [J. Algebra 177 (1995) 164–185], Broué and Malle [Astérisque 212 (1993) 119–189]. In this paper we consider modular representation theory for these algebras in the case where r=p. We assume that the field K contains a primitive pth root of unity . Our method is to study the restrictions of Specht modules for Hecke algebras of type G(p,1,n) with parameters (q;1, ,…, p−1). Suppose that f(q, )≠0 in K (see 4.7 for definition of f(q, )). For any multipartition λ=(λ(1),…,λ(p)) of n, we prove that for any 1 k p−1, where λ[k]=(λ(k+1),λ(k+2),…,λ(p),λ(1),λ(2),…,λ(k)); and if k is the smallest positive integer such that λ=λ[k] (hence k p), we explicitly decompose into a direct sum of p/k smaller -submodules with the same dimensions. As a result, when f(q, )≠0 in K, we show that is split over K and get a complete classification of all the absolutely irreducible -modules. This generalizes earlier work of [C. Pallikaros, J. Algebra 169 (1994) 20–48] and [J. Hu, Manuscripta Math. 108 (2002) 409–430] on Hecke algebras of type Dn (which are included as a special case of our main results).
