The functor of rational representations for p-groups

Abstract

Let k be a field of characteristic p>0. Let C p,k be the category whose objects are the finite p-groups, morphisms are the k-linear combinations of bisets, and composition of morphisms is obtained by k-linear extension from the usual product of bisets. Let F p,k denote the category of k-linear functors from C p,k to the category of k-vector spaces. This paper investigates the structure of the functor kR Q mapping a p-group P to kR Q (P)=k⊗ Z R Q (P) , as an object of F p,k , where R Q (P) is the Grothendieck group of the category of finite dimensional QP -modules. The main result is an explicit description of the lattice of all subfunctors of kR Q . In particular for p odd, it is shown that kR Q is a uniserial object in F p,k . For p=2, the lattice of subfunctors of kR Q can be described as the lattice of closed subsets of a graph whose vertices are the 2-groups of normal 2-rank 1. In both cases a composition series of kR Q is obtained, which leads to a formula giving the dimension of the evaluations S Q, k ( P) of the simple objects S Q, k of F p,k associated to p-groups Q of normal p-rank 1, different from C p . This formula can be phrased in terms of rational representations of P, but also in terms of the geometry of the lattice of subgroups of P, using the notions of basic subgroups and origins. For example, if p=2, the dimension of S 1 ,k (P) is equal to the number of absolutely irreducible QP -modules.