On the character ring of a finite group

Abstract

Let G be a flnite group and let k be a su-ciently large flnite fleld. Let R(G) denote the character ring of G (i.e. the Grothendieck ring of the category of CG-modules). We study the structure and the representations of the commutative algebra k ›Z R(G). Contents Let G be a flnite group. We denote by R(G) the Grothendieck ring of the category of CG- modules (it is usually called the character ring of G). It is a natural question to try to recover properties of G from the knowledge of R(G). It is clear that two flnite groups having the same character table have the same Grothendieck rings and it is a Theorem of Saksonov (S) that the converse also holds. So the problem is reduced to an intensively studied question in character theory: recover properties of the group through properties of its character table. In this paper, we study the k-algebra kR(G) = k ›Z R(G), where k is a splitting fleld for G of positive characteristic p. It is clear that the knowledge of kR(G) is a much weaker information than the knowledge of R(G). The aim of this paper is to gather results on the representation theory of the algebra kR(G): although most of the results are certainy well-known, we have not found any general treatment of these questions. The blocks of kR(G) are local algebras which are parametrized by conjugacy classes of p-regular elements of G. So the simple kR(G)-modules are parametrized by conjugacy classes of p-regular elements of G. Moreover, the dimension of the projective cover of the simple module associated to the conjugacy class of the p-regular element g 2 G is equal to the number of conjugacy classes of p-elements in the centralizer CG(g). We also prove that the radical of kR(G) is the kernel of the decomposition map kR(G) ! k ›Z R(kG), where R(kG) is the Grothendieck ring of the category of kG-modules (i.e. the ring of virtual Brauer characters of G).