Tensor induction for M-Burnside rings

Abstract

Let G be a finite group and H a subgroup. The M-Burnside ring ΩM(H) arising from a monoid functor M for G is a generalization of the Burnside ring Ω(H). We present a certain hypothesis on M such that if M satisfies the hypothesis, then there is a multiplicative map θHG:ΩM(H)→ΩM(G) derived from the tensor induction jndHG:Ω(H)→Ω(G). Let A be a finite abelian G-group. For the monoid functor HA assigning each K≤G the 1st cohomology group H1(K,A) of K with coefficients in A, the HA-Burnside ring ΩHA(H) is isomorphic to the ring of monomial representations of H with coefficients in A. We show that HA satisfies the hypothesis and the associated multiplicative map defines the known tensor (or multiplicative) induction for the rings of monomial representations. Let S be a finite commutative G-monoid. The monoid functor CS assigning each K≤G the set of K-invariants in S satisfies the hypothesis. To extend this fact, we introduce the concept of hereditary monoid functors which satisfy the hypothesis. The monoid functor MS∘ assigning each K≤G the set of normal subgroups of K is a hereditary monoid functor for G on the subgroup lattice of G. In short, we present tensor inductions for HA-, CS-, and MS∘-Burnside rings derived from the tensor induction for Burnside rings.