相关论文: On the unit group of the Burnside ring for some so…

In this note I describe the structure of the biset functor $B^\times$ sending a $p$-group $P$ to the group of units of its Burnside ring $B(P)$. In particular, I show that $B^\times$ is a rational biset functor. It follows that if $P$ is a…

We introduce the theory of biset functors defined on finite categories. Previously, biset functors have been defined on groups, and in that context they are closely related to Mackey functors. Standard examples on groups include…

Let $B^\times$ be the biset functor over $\mathbb{F}_2$ sending a finite group~$G$ to the group $B^\times(G)$ of units of its Burnside ring $B(G)$, and let $\widehat{B^\times}$ be its dual functor. The main theorem of this paper gives a…

Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all finite groups $G$, and let $\mathbb{K}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in $A$. In this…

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…

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…

We generalize Bouc's construction of orthogonal idempotents in the double Burnside algebra to the setting of the double $\mathbb{C}^\times$-fibered Burnside algebra. This yields a structural decomposition of the evaluations of…

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…

In this note we present an algorithm for the construction of the unit group of the Burnside ring $\Omega(G)$ of a finite group $G$ from a list of representatives of the conjugacy classes of subgroups of G.

We introduce the concept of lattice Burnside ring for a finite group G associated to a family of nonempty sublattices of a finite G-lattice assigned to subgroups of G. The slice Burnside ring introduced by S. Bouc is isomorphic to a lattice…

We investigate the structure of the monomial Burnside biset functor over a field of characteristic zero, with particular focus on its restriction kernels. For each finite \( p \)-group \( G \), we give an explicit description of the…

We give a constructive proof of the theorem of Bass and Milnor saying that if $G$ is a finite abelian group then the Bass units of the integral group ring $\Z G$ generate a subgroup of finite index in its units group $\U(\Z G)$. Our proof…

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…

Let $p$ be a prime number, let $H$ be a finite $p$-group, and let $\mathbb{F}$ be a field of characteristic 0, considered as a trivial $\mathbb{F} \mathrm{Out}(H)$-module. The main result of this paper gives the dimension of the evaluation…

Given a finite group $G$, its double Burnside ring $B(G,G)$, has a natural duality operation that arises from considering opposite $(G,G)$-bisets. In this article, we systematically study the subgroup of units of $B(G,G)$, where elements…

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\Z G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4…

We determine the composition factors of the $A$-fibered Burnside functor $kB^A$ for $p$-groups over a field $k$ of characteristic $q$ with $q\neq p$ and cyclic fiber group $A$. We also show that, in this case, $kB^A$ is uniserial.

To the best of our knowledge, there is no explicit, constructive description of the generating set for the unit group $A(G)^\times$ of the Burnside ring associated with a finite group $G$. We resolve this long-standing open question,…

We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. In…

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…