Burnside rings of fusion systems and their unit groups

For a saturated fusion system {\mathcal{F}} on a p-group S, we study the Burnside ring of the fusion system {B(\mathcal{F})}, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring {B(S)}. We give criteria for an element of {B(S)} to be in {B(\mathcal{F})} determined by the {\mathcal{F}}-automorphism groups of essential subgroups of S. When {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p-group, we show that the restriction of {B(G)} to {B(S)} has image equal to {B(\mathcal{F})}. We also show that, for {p=2}, we can gain information about the fusion system by studying the unit group {B(\mathcal{F})^{\times}}. When S is abelian, we completely determine this unit group.

Deflation and tensor induction on the Frobenius–Wielandt morphism

We explore conditions for the Frobenius–Wielandt morphism to commute with the operations of deflation and tensor induction on the Burnside ring. In doing this, we review the commutativity with induction. The techniques used for induction and tensor induction no longer work for deflation, so, in this case, we make use of tools coming from the theory of biset functors.

Burnside rings for Real 2-representation theory: The linear theory

This paper is a fundamental study of the Real 2-representation theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real 2-representations as a Real variant of the Burnside ring of the fundamental group of the 2-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to 2-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real 2-representation theory of 2-groups, as studied by the second author.