Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal p-sylow subgroup

Abstract

Through recent investigations of J. Thrvenaz and P.J. Webb [4-8], the concept of Mackey functors, being originally introduced by J. A. Green [3] and studied by A.W.M. Dress, T. Yoshida, H. Sasaki, and others, has regained considerable interest. An important reason for this is ~i new interpretation of a conjecture of Aiperin in terms of Mackey functors in [8] using results of [7]. Given a commutative ring ,4, by its definition, it is immediate that the category of Mackey functors over .4 for a finite group G is an abelian category, even that it is the category of ,4-representations of a quiver with relations [8]. Namely a Mackey functor for G associates to each subgroup of G a ,(-module and moreover certain /'-linear maps between these spaces satisfying certain relations involving G. Thus a Mackey functor for G can be interpreted as a module over a path algebra of a quiver with relations, the so-called Mackey algebra [6, 9]. The main difference to the usual representation theory of quivers, and thus the main problem of interpreting Mackey functors as such representations, lies in the fact that the maps between these spaces do not lie in the radical of this path algebra, and this path algebra is not basic in general. So it is a natural problem to compute the quiver in the usual sense for the category of Mackey functors for a group G in particular in the case where ,4 is a field of characteristic p dividing the order of G. The question of when there are up to isomorphism only finitely many indecomposable Mackey functors for G led to recent investigations of Thrvenaz and Webb solving this problem in general, and it also led to the investigations we present in this paper which presents in a special situation, namely if G has a normal cyclic p-Sylow subgroup, an explicit computation of the quiver of the now so-called Mackcy algebra. By definition, the Mackey algebra is the basic ,(-algebra whose category of finitely generated