Related papers: A ghost ring for the left-free double Burnside rin…

For a finite group $G$, we introduce a multiplication on the $\QQ$-vector space with basis $\scrS_{G\times G}$, the set of subgroups of $G\times G$. The resulting $\QQ$-algebra $\Atilde$ can be considered as a ghost algebra for the double…

This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of…

Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we…

The double Burnside $R$-algebra $\text{B}_R(G,G)$ of a finite group $G$ with coefficients in a commutative ring $R$ has been introduced by S. Bouc. It is $R$-linearly generated by finite $(G,G)$-bisets, modulo a relation identifying…

This paper develops links between the Burnside ring of a finite group $G$ and the slice Burnside ring}. The goal is to gain a better understanding of ghost maps, idempotents, prime spectrum of these Burnside rings and connections between…

Let R be a (unital) commutative ring, and G be a finite group with order invertible in R. We introduce new idempotents in the double Burnside algebra RB(G,G), indexed by conjugacy classes of minimal sections of G, i.e. pairs (T,S) of…

The bifree double Burnside ring $B^\Delta(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B^\Delta_\circ(G,G)$ of orthogonal units in $B^\Delta(G,G)$. It is shown that this group is always finite and contains…

In this note an `extended Burnside ring' is defined, generated by classes of semisimple module categories over Rep(G) with quasifibre functors. Here G is a finite group and representations are taken over an algebraically closed field of…

(Talk presented at the 1992 ICTP summer workshop in high energy physics and cosmology) The BRST cohomology ring for $(p,q)$ models coupled to gravity is discussed. In addition to the generators of the ghost number zero ring, the existence…

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This…

We explore the concept of conjugation between subgroupoids, providing several characterizations of the conjugacy relation (Theorem A in {\S}1.2). We show that two finite groupoid-sets, over a locally strongly finite groupoid, are…

Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an…

We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra $H$ associated to the factorization of a finite group into two subgroups. The representations of the quantum double…

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 G-stable if it has isomorphic restrictions to G-conjugate subgroups of S.…

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for…

If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map from the Burnside ring to the representation ring of G has a kernel. Its elements are…

We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups H_1 and H_2, we study the tensor product of the associated representation rings R(H_1) and R(H_2)…

Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent $A(G)$ of finite $p$-groups was obtained in a previous article of the first-named author. In this paper, we compute $A(G)$ for any…

Let $G$ be a finite group and $A(G)$ its Burnside ring. For $H \subset G$ let $\mathbb{Z}_H$ denote the $A(G)$-module corresponding to the mark homomorphism associated to $H$. When the order of $G$ is square-free we give a complete…

We extend the notion of a {$p$-permutation equivalence} between two $p$-blocks $A$ and $B$ of finite groups $G$ and $H$, from the definition in [Boltje-Xu 2008] to a virtual $p$-permutation bimodule whose components have twisted diagonal…