The University of Arizona

Bisets and the Double Burnside Algebra of a Finite Group

Bisets and the Double Burnside Algebra of a Finite Group

Series: Algebra and Number Theory Seminar
Location: ENR2 S395
Presenter: Götz Pfeiffer, National University of Ireland, Galway

The double Burnside group $B(G, H)$ of two finite groups $G, H$ is the Grothendieck group of the category of finite $(G, H)$-bisets. Certain bisets encode relationships between the representation theories of $G$ and $H$.  Bouc's biset category provides a framework for studying such relationships, it has finite groups as objects, and $B(G, H)$ as morphisms between $G$ and $H$, with composition induced by the tensor product of bisets.  The endomorphism ring $B(G, G)$ is called the double Burnside ring of $G$.  In contrast to the (ordinary) Burnside ring $B(G)$, the double Burnside ring $B(G, G)$ of a nontrivial group $G$ is not commutative.  In general, little more is known about the structure of $B(G, G)$.

In this talk I'll describe the role of the double Burnside ring in the wider context of the representation theory of finite groups.  In terms of structure, I'll discuss a relatively small faithful matrix representation of the rational double Burnside algebra $\mathbb{Q}B(G,G)$ for certain finite groups $G,$ based on a recent decomposition of the table of marks of the direct product $G \times G$, exhibiting the cellular structure of the algebra $\mathbb{Q}B(G, G)$.  This is joint work with Sejong Park.

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