The abelian monoid of fusion-stable finite sets is free

Abstract

Abstract. We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called Gstable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable Ssets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat non-obvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a saturated fusion system into its associated ghost ring.

Transfer and characteristic idempotents for saturated fusion systems

“…We also recall the Burnside ring of a saturated fusion system F, in the sense of [15], which has a similar mark homomorphism and ghost ring.…”

“…A proof of this claim can be found in [10,Proposition 3.2.3] or [15]. We shall primarily use (ii) and (iii) to characterize F-stability.…”

“…Equivalently, we can consider the actual S-sets that are F-stable: The F-stable sets form a semiring, and we define A(F) to be the Grothendieck group hereof. These two constructions give rise to the same ring A(F) -see [15,Proposition 4.4]. As is the case for the Burnside ring of a group, A(F) has an additive basis, where the basis elements are in one-to-one correspondence with the F-conjugacy classes of subgroups in S.…”

“…Using this stabilization homomorphism, we give a new basis for A(F) (p) . It was shown in [15] that the irreducible F-stable sets form a basis for A(F), but very little is known about their actual structure. The new basis for A(F) (p) , though it only exists after p-localization, is easily described in terms of the homomorphism of marks.…”

“…Such a procedure was used by Broto, Levi and Oliver in [5] to show that every saturated fusion system has at least one "characteristic biset," a set with left and right S-actions satisfying properties suggested by Linckelmann and Webb. A similar procedure was used in [15], to construct all irreducible F-stable S-sets. Both constructions follow the same general idea: To begin with, we are given a finite S-set X (or in general an element of the Burnside ring).…”