Simple 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 homological properties of strict polynomial functors of degree p

We study the homological algebra in the category Pp of strict polynomial functors of degree p over a field of positive characteristic p. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category Pp has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of Pp as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p-weight 1 in Pe for e>p.

Diagonal p-permutation functors, semisimplicity, and functorial equivalence of blocks

Let k be an algebraically closed field of characteristic p>0, let R be a commutative ring, and let F be an algebraically closed field of characteristic 0. We consider the R-linear category FRppkΔ of diagonal p-permutation functors over R. We first show that the category FFppkΔ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations.Next, to any pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor RTG,bΔ in FRppkΔ. We find the decomposition of the functor FTG,bΔ as a direct sum of simple functors in FFppkΔ. This leads to a characterization of nilpotent blocks in terms of their associated functors in FFppkΔ.Finally, for such pairs (G,b) of a finite group and a block idempotent, we introduce the notion of functorial equivalence over R, which (in the case R=Z) is slightly weaker than p-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite p-group D, there is only a finite number of pairs (G,b), where G is a finite group and b a block idempotent of kG with defect isomorphic to D, up to functorial equivalence over F.

Diagonal p-permutation functors

Let k be an algebraically closed field of positive characteristic p, and let F be an algebraically closed field of characteristic 0. We consider the F-linear category FppkΔ of finite groups, in which the set of morphisms from G to H is the F-linear extension FTΔ(H,G) of the Grothendieck group TΔ(H,G) of p-permutation (kH,kG)-bimodules with (twisted) diagonal vertices. The F-linear functors from FppkΔ to F-Mod are called diagonal p-permutation functors. They form an abelian category FppkΔ.We study in particular the functor FTΔ sending a finite group G to the Grothendieck group FT(G) of p-permutation kG-modules, and show that FTΔ is a semisimple object of FppkΔ, equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs (P,s) of a finite p-group P and a generator s of a p′-subgroup acting faithfully on P. This leads to a precise description of the evaluations of these simple functors. In particular, we show that the simple functor indexed by the trivial pair (1,1) is isomorphic to the functor sending a finite group G to FK0(kG), where K0(kG) is the Grothendieck group of projective kG-modules.

Correspondence functors and lattices

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. A main tool for this study is the construction of a correspondence functor associated to any finite lattice T. We prove for instance that this functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple functors. The special case of total orders yields some more specific and complete results.

Simple Functors of Admissible Linear Categories

Generalizing an idea used by Bouc, Thevenaz, Webb and others, we introduce the notion of an admissible R-linear category for a commutative unital ring R. Given an R-linear category \(\mathcal {L}\), we define an \(\mathcal {L}\)-functor to be a functor from \(\mathcal {L}\) to the category of R-modules. In the case where \(\mathcal {L}\) is admissible, we establish a bijective correspondence between the isomorphism classes of simple functors and the equivalence classes of pairs (G, V) where G is an object and V is a module of a certain quotient of the endomorphism algebra of G. Here, two pairs (F, U) and (G, V) are equivalent provided there exists an isomorphism F ← G effecting transport to U from V. We apply this to the category of finite abelian p-groups and to a class of subcategories of the biset category.

Blocks of Mackey categories

For a suitable small category F of homomorphisms between finite groups, we introduce two subcategories of the biset category, namely, the deflation Mackey category MF← and the inflation Mackey category MF→. Let G be the subcategory of F consisting of the injective homomorphisms. We shall show that, for a field K of characteristic zero, the K-linear category KMG=KMG←=KMG→ has a semisimplicity property and, in particular, every block of KMG owns a unique simple functor up to isomorphism. On the other hand, we shall show that, when F is equivalent to the category of finite groups, the K-linear categories KMF← and KMF→ each have a unique block.

The extension algebra of some cohomological Mackey functors

Let k be a field of characteristic p. We construct a new inflation functor for cohomological Mackey functors for finite groups over k. Using this inflation functor, we give an explicit presentation of the graded algebra of self-extensions of the simple functor S1G, when p is odd and G is an elementary abelian p-group.

Matrix factorisations for rational boundary conditions by defect fusion

A large class of two-dimensional $\mathcal{N}=(2,2)$ superconformal field theories can be understood as IR fixed-points of Landau-Ginzburg models. In particular, there are rational conformal field theories that also have a Landau-Ginzburg description. To understand better the relation between the structures in the rational conformal field theory and in the Landau-Ginzburg theory, we investigate how rational B-type boundary conditions are realised as matrix factorisations in the $SU(3)/U(2)$ Grassmannian Kazama-Suzuki model. As a tool to generate the matrix factorisations we make use of a particular interface between the Kazama-Suzuki model and products of minimal models, whose fusion can be realised as a simple functor on ring modules. This allows us to formulate a proposal for all matrix factorisations corresponding to rational boundary conditions in the $SU(3)/U(2)$ model.

Vanishing evaluations of simple functors

The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on G.

Simple biset functors and double Burnside ring

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 introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(−,H) by the radical of the bilinear form is a semi-simple functor. This allows for a description of the evaluation of simple functors, hence of simple modules for the double Burnside ring.

Simplicial descent categories

Let D be a category and E a class of morphisms in D . In this paper we study the question of how to transfer homotopic structure from the category of simplicial objects in D , Δ ∘ D , to D through a ‘good’ functor s : Δ ∘ D → D , which we call simple functor. For instance, the Bousfield–Kan homotopy colimit in a Quillen simplicial model category is a good simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on Δ ∘ D . We use these facts to produce cofiber sequences on the localized category of D by E , which give rise to a natural Verdier triangulated structure in the stable case.

Composition factors of functors

Let X be a family of finite groups satisfying certain conditions and K be a field. We study composition factors, radicals, and socles of biset and related functors defined on X over K . For such a functor M and for a group H in X , we construct bijections between some classes of maximal (respectively, simple) subfunctors of M and some classes of maximal (respectively, simple) K Out ( H ) -submodules of M ( H ) . We use these bijections to relate the multiplicity of a simple functor S H , V in M to the multiplicity of V in a certain K Out ( H ) -module related to M ( H ) . We then use these general results to study the structure of one of the important biset and related functors, namely the Burnside functor B K which assigns to each group G in X its Burnside algebra B K ( G ) = K ⊗ Z B ( G ) where B ( G ) is the Burnside ring of G. We find the radical and the socle of B K in most cases of X and K . For example, if K is of characteristic p > 0 and X is a family of finite abelian p-groups, we find the radical and the socle series of B K considered as a biset functor on X over K . We finally study restrictions of functors to nonfull subcategories. For example, we find some conditions forcing a simple deflation functor to remain simple as a Mackey functor. For an inflation functor M defined on abelian groups over a field of characteristic zero, we also obtain a criterion for M to be semisimple, in terms of the images of inflation and induction maps on M.

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

AbstractWe describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

The Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

In set theory the cardinality of the continuum \(|{\mathbb R}|\) is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations \(k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}\), taking values on odd order groups over the field k of characteristic 2, is equal to \(|{\mathbb R}|\). When the characteristic q > 0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations SC,k(G), of the simple functor SC,k, at any group G of order prime to q and being associated to a suitable cyclic group C.

Complexity and cohomology of cohomological Mackey functors

Let k be a field of characteristic p > 0 . Call a finite group G a poco group over k if any finitely generated cohomological Mackey functor for G over k has polynomial growth. The main result of this paper is that G is a poco group over k if and only if the Sylow p-subgroups of G are cyclic, when p > 2 , or have sectional rank at most 2, when p = 2 . A major step in the proof is the case where G is an elementary abelian p-group. In particular, when p = 2 , all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor S 1 G , by explicit generators and relations.

Extension groups between simple Mackey functors

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.

RESTRICTING GLOBALLY-DEFINED MACKEY FUNCTORS TO NILPOTENT FINITE GROUPS

Let B be the Burnside ring considered as a globally-defined Mackey functor, and k be a field of positive characteristic q. We prove that, when k ⊗ℤ B is restricted to nilpotent groups of order prime to q, the only simple Mackey functors that can appear as subquotients of it are of the form SE,k, where E is the direct product of elementary abelian p-groups. In the special case k = 𝔽2, the simple Mackey functor SE,𝔽2 must appear as a filtration factor in 𝔽2 ⊗ℤ B for every direct product of elementary abelian p-groups E of odd order.

Sur des foncteurs simples

Let k be a field of positive characteristic p. One considers some categories, whose objects are given classes of finite p-groups, and morphisms are given classes of k-virtual bisets, i.e., linear combinations of bisets with coefficients in k. The category of k-linear functors from such a category to the category of k-vector spaces is abelian, and one can try to classify and describe its simple objects, or its projective objects. By specific subfunctors of the Burnside functor, which have a unique simple quotient S Q,k , one will get some estimates on the k-dimension of the evaluations of these simple functors. These evaluations are equalities for abelian p-groups, and for such groups P the result is even stronger, since it provides some explicit k-bases for the evaluations S Q,k (P).

The group of endo-permutation modules

The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of Q\otimes D(-) viewed as a functor, which turns out to be a simple functor S_{E,Q}, indexed by the elementary group E of order p^2 and the trivial Out(E)-module Q. Finally we describe a rather strange exact sequence relating Q\otimes D(P), Q\otimes B(P), and Q\otimes R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of QP-modules.