Perfect isometries and Murnaghan-Nakayama rules

Abstract

This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two p-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for p-blocks of alternating groups (where the blocks must also have the same sign when p is odd), of double covers of alternating and symmetric groups (for p odd, and where we obtain crossover isometries when the blocks have opposite signs), of complex reflection groups G(d, 1, n) (for d prime to p), of Weyl groups of type B and D (for p odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks, in a way which should be of independent interest.

Basic Sets for the Double Covering Groups of the Symmetric and Alternating Groups in Odd Characteristic

“…This proves that the set of spin characters of G is a union of C-blocks (see [7,Prop 2.14] and [23, §1]). By Lemma 4.1 and Proposition 4.2 of [9], the set of irreducible characters of G with P in their kernel is a C-basic set of G. The intersection of this basic set with the set of spin characters of G, which is given by Proposition 3.2, is thus a C-basic set for the set of spin characters of G.…”

“…The second author was supported by the EPSRC grant Combinatorial Representation Theory EP/M019292/1. In [7] (see also [23]), the authors gave a notion of perfect isometries, which generalizes to C the perfect isometries introduced by Broué in [5]. Such a perfect isometry I furthermore satisfies I( χ) = I(χ) for all χ ∈ B, whence in particular sends a basic set to a basic set.…”

“…In Section 4, we adapt the perfect isometry constructed by Livesey in [24], where he proves Broué's Abelian Defect Perfect Isometry Conjecture, and we apply the methods and results of [7]. This produces, for any p-block B of a double covering group of the symmetric or alternating groups, a perfect isometry between B and a C-block of G or a D-block of H, from which Theorem 1.1 is deduced.…”

Perfect isometries between blocks of complex reflection groups

“…η ′ ) = 0. Since J q,γ,γ ′ ,w is a perfect isometry (by [2,Theorem 5.4]), this implies that g η ′ are both p-regular or both p-singular, which, by the above, shows that g η and g η ′ , and thus x and x ′ , are both p-regular or both p-singular. Hence Property (2) of Definition 4.6 holds.…”

“…where, for any q dividing e (and such that γ/q, γ ′ /q and w/q are defined), J q,γ,γ ′ ,w is the perfect isometry described in [2,Theorem 5.4] between the p-block β of G(de/q, 1, r/q) with p-core γ/q and p-weight w/q and the p-block β ′ of G(de/q, 1, r ′ /q) with p-core γ ′ /q and (same) p-weight w/q.…”