Related papers: $p$-Biset Functor of Monomial Burnside Rings

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…

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…

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.

We introduce and study the category of $p$-bifree biset functors for a fixed prime $p$, defined via bisets whose left and right stabilizers are $p'$-groups. This category naturally lies between the classical biset functors and the diagonal…

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…

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…

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 theory of bisets has been very useful in progress towards settling the longstanding question of determining units for the Burnside ring. In 2006 Bouc used bisets to settle the question for $p$-groups. In this paper, we provide a…

This note has two purposes: First, to present a counterexample to a conjecture parametrizing the simple modules over Green biset functors, appearing in an author's previous article. This parametrization fails for the monomial Burnside ring…

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…

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…

In this paper, I give several characterizations of {\em rational biset functors over $p$-groups}, which are independent of the knowledge of genetic bases for $p$-groups. I also introduce a construction of new biset functors from known ones,…

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…

We determine the structure of the fibered biset functor sending a finite group $G$ to the complex vector space of complex valued class functions of $G$. Previously, it is studied as a biset functor by Bouc and as a $\mathbb…

We classify simple modules over the Green biset functor of section Burnside rings.

In the previous article 'A Mackey-functor theoretic interpretation of biset functors', we have constructed the 2-category $\mathbb{S}$ of finite sets with variable finite group actions, in which bicoproducts and bipullbacks exist. As shown…

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to…

Let G be a finite group and K be a field of characteristic zero. Our purpose is to investigate the ideals of the slice Burnside functor K{\Xi}. It turns out that they are the subfunctors F of K{\Xi} such that for any finite group G, the…

For all subgroups $H$ of a cyclic $p$-group $G$ we define norm functors that build a $G$-Mackey functor from an $H$-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the…

We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an…