Biset Functors Research Articles

Simple functors over the Green biset functor of section Burnside rings

Green biset functors are introduced by Serge Bouc as a general framework to work with biset functors having an additional ring structure. Well-known examples include the functors of (fibered) Burnside rings, trivial source rings and character rings. Serge Bouc also introduced two generalizations of Burnside rings, the so-called slice and section Burnside rings which are also Green biset functors. In this paper, we classify simple modules over the Green biset functor of section Burnside rings.

On the essential algebra of the shifted Burnside biset functor

We describe the essential algebra, kBTˆ(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 this case, kBTˆ(G) is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of G×G×T. The second case is when G and T are any finite groups satisfying (|G|,|T|)=1 and k is a commutative unitary ring. In this case, kBTˆ(G) is isomorphic to a semidirect product of Out(G) and kBZ(G)(T), the monomial Burnside ring of T with coefficients in Z(G).The aim of the article is to consider the natural set of generators of kBTˆ(G) coming from the transitive elements in kBT(G×G) and explore some cases in which it is possible to give a basis for kBTˆ(G) in this set.

On the image of the trivial source ring in the ring of virtual characters of a finite group

We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of $p$-rational complex characters. We use Boltje and Co\c{s}kun's theory of fibered biset functors to determine the structure of the cokernel. An essential tool in the determination of this structure is Bouc's theory of rational $p$-biset functors.

Green fields

We introduce Green fields, as commutative Green biset functors with no non-trivial ideals. We state some of their properties and give examples of known Green biset functors which are Green fields. Among the properties, we prove some criterions ensuring that a Green field is semisimple. Finally, we describe a type of Green field for which its category of modules is equivalent to a category of vector spaces over a field.

Shifted functors of linear representations are semisimple

We prove that, for any fields $k$ and $\mathbb{F}$ of characteristic $0$ and any finite group $T$, the category of modules over the shifted Green biset functor $(kR_{\mathbb{F}})_T$ is semisimple.

A Functorial Presentation of Units of Burnside Rings

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 characterization of the cokernel of the natural injection from \(B^\times \) in the dual Burnside functor \(\widehat{\mathbb {F}_2B}\), or equivalently, an explicit set of generators \(\mathcal {G}_\mathcal {S}\) of the kernel L of the natural surjection \(\mathbb {F}_2B\rightarrow \widehat{B^\times }\). This yields a two terms projective resolution of \(\widehat{B^\times }\), leading to some information on the extension functors \(\mathrm{Ext}^1(-,B^\times )\). For a finite group G, this also allows for a description of \(B^\times (G)\) as a limit of groups \(B^\times (T/S)\) over sections (T, S) of G such that T/S is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor \(B^\times \) is not finitely generated, and that its dual \(\widehat{B^\times }\) is finitely generated, but not finitely presented. The last result of the paper shows in addition that \(\mathcal {G}_\mathcal {S}\) is a minimal set of generators of L, and it follows that the lattice of subfunctors of L is uncountable.

The A-fibered Burnside Ring as A-Fibered Biset Functor in Characteristic Zero

Let A be an abelian group such that torn(A) is finite for every n ≥ 1 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 paper we prove fundamental properties of the A-fibered Burnside ring functor $B_{\mathbb{K}}^{A}$ as an A-fibered biset functor over K. This includes a description of the composition factors of $B^{A}_{\mathbb{K}}$ and the lattice of subfunctors of $B_{\mathbb{K}}^{A}$ in terms of what we call BA-pairs and a poset structure on their isomorphism classes. Unfortunately, we are not able to classify BA-pairs. The results of the paper extend results of Coskun and Yilmaz for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor as a biset functor over fields of characteristic zero. In the latter case, BA-pairs become Bouc’s B-groups which are also not known in general.

Essential support of Green biset functors via morphisms

In this note, we introduce a criterion to detect vanishing of essential algebras of a Green biset functor by means of morphisms. We introduce the inflation morphism and restriction morphism to prove that the essential supports of a Green biset functor and its shifted functors are the same. We use the kernel of the restriction to give a characterization of the seeds of the shifted functors.

The functor of complex characters of finite groups

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 C×-fibered biset functor by Boltje and the second author. In this paper, we complete the study by considering proper non-trivial subgroups of C× as the fiber group. As a corollary, we obtain a (previously known) set of composition factors of the biset functor of p-permutation modules.

Deflation and tensor induction on the Frobenius–Wielandt morphism

Abstract 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.

The center of a Green biset functor

For a Green biset functor $A$, we define the commutant and the center of $A$ and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of $A$-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group.

Idempotents in $$F_+(G)$$

In this paper, we compute the idempotents of the ring $$F_+(G)$$ as defined by Boltje (J Algebra 206:293–343, 1998) in the particular case when the Green biset functor F is such that for all subgroups H of a finite group G, F(H) is a torsion-free ring, finitely-generated as an Abelian group, and has only the trivial idempotents. In this case, the only idempotents in $$F_+(G)$$ are those arising from the Burnside ring B(G).

Obstructions for gluing biset functors

We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the subquotients of a finite group G. The obstruction groups for this theory are the reduced cohomology groups of a category DG⁎ whose objects are the sections (U,V) of G, where 1≠V⊴U≤G, and whose morphisms are defined as a generalization of morphisms in the orbit category. Using this obstruction theory, we calculate the obstruction group for some well-known p-biset functors, such as the Dade group functor defined on p-groups with p odd.

Around evaluations of biset functors

For a non-vanishing group, we show that the evaluation functor induces an equivalence between the category of modules over the double Burnside algebra and a certain category of biset functors. Using this equivalence, we deduce that over a field of characteristic zero, the double Burnside algebra of a non-vanishing group is a quasi-hereditary algebra. We also show that the double Burnside algebra over a field is self-injective if and only if it is semisimple.

K-theory, genotypes, and biset functors

Let p be an odd prime number. In this paper, we show that the genome$$\Gamma (P)$$ of a finite p-group P, defined as the direct product of the genotypes of all rational irreducible representations of P, can be recovered from the first group of K-theory $$K_1(\mathbb {Q}P)$$. It follows that the assignment $$P\mapsto \Gamma (P)$$ is a p-biset functor. We give an explicit formula for the action of bisets on $$\Gamma $$, in terms of generalized transfers associated to left free bisets. Finally, we show that $$\Gamma $$ is a rational p-biset functor, i.e. that $$\Gamma $$ factors through the Roquette category of finite p-groups.

The −+ and −+ constructions for biset functors

In this article we define the −+-construction and the −+-construction, that was crucial in the theory of canonical induction formulas (see [3]), in the setting of biset functors, thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors. Additionally, this provides a unified approach to the study of a class of functors including the Burnside ring, the monomial Burnside ring and global representation ring.

On the ideals and essential algebras of shifted functors of linear representations

We present a study on the shifted Green biset functors kRF,G of linear F-representations with coefficients over k, for fields k and F of characteristic zero and a finite group G. We provide a criterion for the vanishing of their essential algebras and we prove that the condition of uniqueness of minimal groups for simple modules holds for these functors. We give a parametrization of a family of simple kRQ,G-modules. We also prove the semisimplicity of the category of CRC,G-modules by means of an equivalence with a category of modules over a semisimple algebra.

Fibered biset functors

The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules. The main result of this paper is the classification of simple fibered biset functors.

On the unit group of the Burnside ring as a biset functor for some solvable groups

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 standard basis for the unit group of the Burnside ring for groups that contain a abelian subgroups of index two. We then extend this result to groups G, where G has a normal subgroup, N, of odd index, such that N contains an abelian subgroups of index 2. Next, we study the structure of the unit group of the Burnside ring as a biset functor, B× on this class of groups and determine its lattice of subfunctors. We then use this to determine the composition factors of B× over this class of groups. Additionally, we give a sufficient condition for when the functor B×, defined on a class of groups closed under subquotients, has uncountably many subfunctors.

Idempotents of double Burnside algebras, L-enriched bisets, and decomposition of p-biset functors

Let R be a (unital) commutative ring, and G be a finite group with order invertible in R. We introduce new idempotents ϵT,SG in the double Burnside algebra RB(G,G) of G over R, indexed by conjugacy classes of minimal sections (T,S) of G (i.e. sections such that S≤Φ(T)). These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, the evaluation F(G) splits as a direct sum of specific R-modules indexed by minimal sections of G, up to conjugation.The restriction of these constructions to the biset category of p-groups, where p is a prime number invertible in R, leads to a decomposition of the category of p-biset functors over R as a direct product of categories FL indexed by atoricp-groups L up to isomorphism.We next introduce the notions of L-enriched biset and L-enriched biset functor for an arbitrary finite group L, and show that for an atoric p-group L, the category FL is equivalent to the category of L-enriched biset functors defined over elementary abelian p-groups.Finally, the notion of vertex of an indecomposable p-biset functor is introduced (when p∈R×), and when R is a field of characteristic different from p, the objects of the category FL are characterized in terms of vertices of their composition factors.