Gluing torsion endo-permutation modules

Abstract

: Let k be a field of characteristic p, and P be a finite p-group, where p is an odd prime. In this paper, we consider the problem of gluing compatible families of endo-permutation modules : being given a torsion element M Q in the Dade group D(N P (Q)/Q), for each non-trivial subgroup Q of P , subject to obvious compatibility conditions, we show that it is always possible to find an element M in the Dade group of P such that Defres P N P (Q)/Q M = M Q for all Q, but that M need not be a torsion element of D(P ). The obstruction to this is controlled by an element in the zero-th cohomology group over F 2 of the poset of elementary abelian subgroups of P of rank at least 2. We also give an example of a similar situation, when M Q is only given for centric subgroups Q of P . Moreover, general results about biset functors and the Dade functor are given in two appendices.

The Dade group of a fusion system

“…The purpose of the present paper is to investigate compatibility issues with respect to an arbitrary fusion system F on a finite p-group P. After a brief review on fusion systems in Section 2, we define in Section 3 the Dade group DðP; FÞ of a fusion system F on a finite p-group P and relate this to the definition of endo-ppermutation modules, due to Urfer [24]. In Sections 4 and 5, we extend some results of Bouc and Thévenaz [8], [9] to this context, and in Section 6, we describe some examples of Dade groups of fusion systems.…”

“…Bouc and Thévenaz proved in [8,Theorem 5.1] that for p an odd prime and P a non-cyclic finite p-group there is a short exact sequence of F 2 -vector spaces…”

“…One of the questions, first raised by Puig in [21], is whether this family can be 'glued together' to an endo-permutation kP-module V such that Defres P R=Q ðV Þ is equal to V Q in the Dade group DðR=QÞ for any section R=Q of P as above. Following [21], this gluing problem has an a‰rmative answer if P is abelian, which in turn is used to show the existence of a stable equivalence of Morita type between the block and its Brauer correspondent under the assumption that the inertial quotient acts freely on the non-trivial elements of P. In [8], Bouc and Thévenaz gave a more general criterion for when the gluing problem has a solution provided that the involved endo-permutation modules are torsion in their respective Dade groups and p is an odd prime. The purpose of the present paper is to investigate compatibility issues with respect to an arbitrary fusion system F on a finite p-group P. After a brief review on fusion systems in Section 2, we define in Section 3 the Dade group DðP; FÞ of a fusion system F on a finite p-group P and relate this to the definition of endo-ppermutation modules, due to Urfer [24].…”

Simple biset functors and double Burnside ring

“…The generalized Mackey formula (see Proposition A1 in [BT1] and Lemma 2.5 in [BT2]) tells us how to decompose the biset Defres G S/T Indinf G J/K as a sum of transitive bisets (using butterflies, as defined in [BT2]). Many terms factor through a smaller subquotient and are therefore zero in kB(G, H).…”

“…One needs the generalized Mackey formula (Proposition A1 in [BT1] and Lemma 2.5 in [BT2]), which tells us how to decompose the biset Defres G S/T Indinf G S/T as a sum of transitive bisets, using butterflies, as defined in [BT2]. The condition that T is contained in the Frattini subgroup Φ(S) implies that e G S Indinf G S/T = Indinf G S/T e S/T S/T and this is used to show that many terms in the the sum lie in fact in I(S/T, S/T ).…”