A characterization of blocks with vertices

Abstract

In modular representation theory, one regards a block B of a finite group G as a G-algebra or a G x G-module. In [6], Green defined G-algebras and generalized defect groups to G-algebras, and proved that the G x G-module B have vertex DA = {(d, d) E G x G: LIE Dj, where D is the defect group of B. After, in [7, Sect. 31, Puig introduced the concept of interior G-algebras which is a special case of G-algebras. An interior G-algebra becomes a G-algebra and a G x G-module. In this paper, we treat mainly interior G-algebras. Let G be a finite group, p a prime number and k a splitting field for G of characteristic p. Let 0 be a complete discrete valuation ring with unique maximal ideal (rr) and the residue field O/(z) is k. In this paper, any Co-algebra A is an O-free module with a finite rank as an O-module, and any A-module is an O-free left A-module with an finite rank as an Comodule. For an O-algebra A and O-algebra homomorphism p of O[G] to A such that p( 1) = 1 A, where 1 A is the identity element of A, the pair (A, p) is called an interior G-algebra. Then the O-algebra A becomes a G x Gmodule defined by (g, h) a = p(g) ap(h ‘), where a is an element of A and (g, h) is an element of G x G. Let V be an Co[G]-module and H be a subgroup of G. A subspace VH consists of the fixed points of V under the action of H. Whenever H’ is a subgroup of G such that H < H’, the trace map Tr;‘: VH + V“’ is defined by v H C gv, where g runs over a complete set of representatives in H’ of H’/H. We denote by Vz’ the image Trz’ (V”). For p-subgroup P of G we denote by V(P) the O-module VP/C Vg + (rc) VP, where Q runs over the family of proper subgroups of P. See [2] or [3]. For an interior G-algebra (A, p) we often apply this notation to the O[G*]-module AIGd, which is the restriction of A to Gd = {(g, g) E G x G: g E G}, and use similar notation AH, Tr:‘, AZ’ and A(P). Then AZ’ is a two-sided ideal in AH’. If AC is a local ring, we call the 344 OOZl-8693/87 $3.00