Canonical induction, Green functors, lefschetz invariant of monomial G-posets

Abstract

Green functors are a kind of group functor, rather like Mackey functors, but with a further multiplicative structure. They are defined on a category whose objects are finite groups and whose morphisms are generated by maps such as induction, restriction, inflation, deflation. The aim of this thesis is general formulation for canonical induction, suitable for Green functors, optionally equipped with inflations. Let p be a prime number. In Section 3, we apply the Boltje’s theory of canonical induction [1] to p-permutation modules and give a restriction-preserving Z[1/p]- linear canonical induction formula from the inflations of projective modules. In Section 4, we give a general formulation of canonical induction theory for Green biset functors equipped with induction, restriction, inflation maps. Let G be a finite group and C be an abelian group. In Section 5, motivated in part by a search for connection with Peter Symonds’ proof [2] of the integrality of a canonical induction formula, we introduce a Lefschetz invariant for the Cmonomial Burnside ring. These invariants let us to construct generalize tensor induction functors associated to any C-monomial (G, H)-biset from the category of C-monomial G-posets to the category of C-monomial H-posets. We will show that these functors induce well-defined tensor induction maps from BC(G) to BC(H), which in turn gives a group homomorphism BC(G) × → BC(H) × between the unit groups of C-monomial Burnside rings.