MODULAR PERMUTATION REPRESENTATIONS ( l )

A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some pte-viously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Btauet's theory of blocks to Green's work on indecomposable modules. The purpose of this article is to present proofs of the results announced in [15]. Statements of these results have been included here, though a number of explanatory remarks and general background references have not been repeated. With the exception of $0, the sections of this paper have been named according to the features of the classical modular theory with which they are most closely related. 0. The centralizer ring. Throughout this paper G is a finite group acting on a finite set Q (perhaps not transitively or faithfully) and p is a fixed prime. If S is any commutative ring with identity, we define the S-centralizer ring V,(C) = V,(G, In) to be the collection of all matrices with entries from S that commute with the permutation matrices determined (with respect to some fixed ordering of Q) by elements of G. In case S is the ring of rational integers, we write only V(G) for VS(G) and refer to V(G) as the centralizer ring. The standard basis matrices {Ai]:= are obtained from the full set {O,\:=, of orbits of G on n x fl by setting the a, p entry of Ai equal to 1 for (a, p) E Oi and 0 otherwise. These matrices always form an S-basis for V,(G). is isomorphic to the tensor product XV(G).