The modular representation algebra of a finite group

Representation algebras 1.1.Notation and terminology.G is a finite group, with unit element e.k is a field of characteristic p.By a G-module M is meant a (/c, G)-module.Elements of G act as right operators on M, and me m (m M).The k-dimension dim M of M is assumed finite.For example, F F(k, G) is the regular G-module, i.e., the group algebra of G over k, regarded as G-module, and ]ca is the unit G-module, i.e., the field ]c, made into a "trivial" G-module, i.e.,Kx K (ek, xeG).For any G-moduleM, {M} is the class of all G-modules isomorphic to M.V (i runs over a suitable index set I) is a set of representatives of the classes {V} of indecomposable G-modules.The number of these indecom- posable classes is finite if and only if either p 0, or p is a finite prime such that the Sylow p-subgroups of G are cyclic (D.G. Higman [5]).F (j 1, n) is a set of representatives of the classes {FA of irreducible G-modules.The number n of these is always finite.If k is algebraically closed, n is equal to the number of p-regular classes of G (R. Brauer, see [1], [2]).If M', M" are G-modules, M' M" denotes their direct sum.If M is a G-module, and s a nonnegative integer, sM denotes the direct sum of s iso- morphic copies of M.1.2.Let c be an arbitrary commutative ring with identity element.Then the representation algebra At(k, G) of the pair (], G), with coefficients in is defined as follows.It is the c-module generated by the set of all isomor- phism classes M} of G-modules, subject to relations {M} for all M, M', M" such that M -M' M', and equipped with the bilinear multiplication given by IM}IM'} {M (R) M'}.Here M (R) M M(R)k M' is made G-module by (m (R) m')x mx (R) m'x (m M, m Mr, x G).By the Krull-Schmidt theorem for G-modules, At(It, G) is free as c-module, and the V} (i e I) form a c-basis.A (], G) is a commutative, associative al- gebra over c, and has identity element 1The Grothendieclc algebra A* (t, G) is the quotient of A (1, G) by the ideal J