Yukio Tsushima

Notes on trivial source modules

Let G be a finite group and k an algebraically closed field of characteristic p. An indecomposable fcG-module with a vertex Q is said to be a weight module if its Green correspondent with respect to (G,Q,NG(Q)) is simple. Let B be a block of kG. Alperin [1] conjectured that the number of the weight modules belonging to B equals that of the simple modules in B. If this is the case and a defect group of B a TI set, then it can be shown under some additional assumption that the socles of weight modules are simple, which in turn determine the isomorphism classes of the weight modules this holds if G is a simple group with a cyclic Sylow /7-subgroup. This rather surprising property has been known to hold for finite groups of Lie type of characteristic p. However little is known about general properties of weight modules. In the final section we shall study solvable groups that have only simple weight modules. Throughout this paper G denotes a finite group and k an algebraically closed field of prime characteristic p. For a fcG-module Λf, hd(M), soc(M) and P(M) denote the head, socle and projective cover of M respectively. If N is a fcG-module, N\M indicates that N is isomorphic to a direct summand of M, and (N,M) denotes the multiplicity of N as a summand of M. We fix a block B of kG and let D be its defect group. IRR(/?) denotes a full set of non-isomorphic simple modules in BJ(B) its cardinality and WM(£) a full set of non-isomorphic weight modules belonging to £.Let / be the Green correspondence with respect (G,A //), where H=NG(D\ If WM(£|D) denotes the subset of WM(£) consisting of the weight modules with vertices D and b the Brauer correspondent of B in kH, then / induces a bijection between WM(£|I>) and IRR(fe). The author thanks the referee for improving the proof of Proposition 4 below.

Young diagrams and simple constituents of the Specht modules

We discuss the simple constituents of Specht module S for the symmetric group Sn defined over the field of p elements. We firstly give an easier proof to the result in [6] which asserts that there exists a simple constituent of S with the shape of “a branch” of (Theorem 3.3), and secondly give a sufficient condition for to have a particular type branch as a constituent (Proposition 3.4).