Some Remarks on a Conjecture of Alperin
نویسندگان
چکیده
Let G be a finite group, p be a prime, k be an algebraically closed field of characteristic p. J. L. Alperin has conjectured that for any finite group G, # (isomorphism types of simple fcG-modules) = £ # (isomorphism types of projective simple A: (C) where the sum is taken over a set of representatives of the conjugacy classes of /7-subgroups of G. There is also a ' blockwise' version of this conjecture which will be discussed in detail later. See also [2]. In this paper, we present some results which may be viewed in several ways. On the one hand, they may be viewed as reformulations of the conjecture. On the other hand, they can be seen as consequences and applications for groups in which the conjecture can be shown to hold. The main idea of the paper is to place the conjecture in another context, that of certain simplicial complexes which have been studied by S. Bouc [4], K. S. Brown [7], D. Quillen [15], J. Thevenaz [16] and P. J. Webb [17].
منابع مشابه
Alperin’s Conjecture for Algebraic Groups
We prove analogues for reductive algebraic groups of some results for finite groups due to Knörr and Robinson from ‘Some remarks on a conjecture of Alperin’, J. London Math. Soc (2) 39 (1989), 48–60, which play a central rôle in their reformulation of Alperin’s conjecture for finite groups.
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