Mackey Algebra Research Articles

Derivations of Mackey algebras

We describe derivations of finitary Mackey algebras over fields of characteristics not equal to $2.$ We prove that an arbitrary derivation of an associative finitary Mackey algebra or one of the Lie algebras $\mathfrak{sl}_{\infty}(V|W)$, $\mathfrak{o}_{\infty}(f)$ is an adjoint operator of an element in the corresponding Mackey algebra. It provides description of derivations of all algebras in the Baranov-Strade classification of finitary simple Lie algebras. The proof is based on N. Jacobson's result on derivations of associative algebras of linear transformations of an infinite-dimensional vector space and the results on Herstein's conjectures.

Automorphisms of Mackey groups

We consider total subspaces of linear functionals on an infinite-dimensional vector space and the related Mackey algebras and groups. We outline the description of automorphisms of Mackey groups SL∞(V|W), O∞(f), and SU∞(f) over fields of characteristics not equal to 2, 3. Moreover, the paper explores the relationship between field automorphisms and automorphisms of the aforementioned groups. J.Hall proved that infinite simple finitary torsion groups are the alternating groups on infinite sets or Mackey groups over a field, which is an algebraic extension of a finite field. J.Schreier and S.Ulam described automorphisms of infinite alternating groups. With the description of automorphisms of finitary Mackey groups and special finitary unitary Mackey groups we finish classification of automorphisms of all infinite simple finitary torsion groups over fields of characteristics not equal to 2, 3. The proof is based of description of automorphisms of elementary linear groups over associative rings that due to I.Golubchik, A.Mikhalev and E.Zelmanov.

Mackey Algebras which are Gorenstein

We complete the picture available in the literature by showing that the integral Mackey algebra is Gorenstein if and only if the group order is square-free, in which case it must have Gorenstein dimension one. We illustrate this result by looking in details at the examples of the cyclic group of order four and the Klein four group.

On Morita and derived equivalences for cohomological Mackey algebras

By results of the second author, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras.

Equivalences between blocks of cohomological Mackey algebras

Let \(G\) be a finite group and \((K,\mathcal {O},k)\) be a \(p\)-modular system which is large enough. Let \(R=\mathcal {O}\) or \(k\). There is a bijection between the blocks of the group algebra \(RG\) and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra \(co\mu _{R}(G)\). Here, we introduce the notion of permeable derived equivalence and we prove that a permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Broué’s abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.

Equivalences between blocks of p-local Mackey algebras

Let G be a finite group and (K,O,k) be a p-modular system. Let R=O or k. There is a bijection between the blocks of the group algebra and the blocks of the so-called p-local Mackey algebra μR1(G). Let b be a block of RG with abelian defect group D. Let b′ be its Brauer correspondant in NG(D). It is conjectured by Broué that the blocks RGb and RNG(D)b′ are derived equivalent. Here we look at equivalences between the corresponding blocks of p-local Mackey algebras. We prove that an analogue of the Broué's conjecture is true for the p-local Mackey algebras in the following cases: for the principal blocks of p-nilpotent groups and for blocks with defect 1.

Trace maps for Mackey algebras

Let G be a finite group and R be a commutative ring. The Mackey algebra μR(G) shares a lot of properties with the group algebra RG however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetry of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra.Using the fact that the category of Mackey functors is a closed symmetric monoidal category, we prove that the Mackey algebra μR(G) is a symmetric algebra if and only if the family of Burnside algebras (RB(H))H⩽G is a family of symmetric algebras with a compatibility condition.As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of G is symmetric if and only if the order of G is square free. Finally, if (K,O,k) is a p-modular system for G, we show that the Mackey algebras μO(G) and μk(G) are symmetric if and only if the Sylow p-subgroups of G are of order 1 or p.

Fused Mackey functors

Let \(G\) be a finite group. In [6], the question of comparing Mackey functors for \(G\) and biset functors defined on subgroups of \(G\) and bifree bisets as morphisms has been considered. This paper proposes a different approach to this problem, from the point of view of various categories of \(G\)-sets. In particular, the category \(G\hbox {-}{\underline{\mathsf{set}}}\) of fused\(G\)-sets is introduced, as well as the category \(\underline{\mathbf {S}}(G)\) of spans in \(G\hbox {-}{\underline{\mathsf{set}}}\). The fused Mackey functors for \(G\) over a commutative ring \(R\) are defined as \(R\)-linear functors from \(R\,\underline{\mathbf {S}}(G)\) to \(R\)-modules. They form an abelian subcategory \(\mathsf {Mack}_R^f(G)\) of the category of Mackey functors for \(G\) over \(R\). The category \(\mathsf {Mack}_\mathbb {Z}^f(G)\) is equivalent to the category of conjugation Mackey functors of [6]. The category \(\mathsf {Mack}_R^f(G)\) is also equivalent to the category of modules over the fused Mackey algebra\(\mu _R^f(G)\), which is a quotient of the usual Mackey algebra \(\mu _R(G)\) of \(G\) over \(R\).

On the Cartan matrix of Mackey algebras

Let k be a field of characteristic p > 0, and let G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra μ k (G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra coμ k (G) of G over k, and a characterization of the groups G for which this matrix is nonsingular. The third result is a generalization of this rank formula and characterization to blocks of coμ k (G): in particular, if b is a block of kG, the Cartan matrix of the corresponding block coμ k (b) of coμ k (G) is nonsingular if and only if b is nilpotent with cyclic defect groups.

Ring of subquotients of a finite group I: Linearization

We introduce the ring Λ ( G ) of subquotients of a finite group G. As an abelian group, it is free on the set of conjugacy classes of subquotients of the group G. The ring Λ ( G ) integrally extends the Burnside ring and there is a linearization map with range the Grothendieck ring of the Mackey algebra.

The socle of a projective Mackey functor for a p-group

We study the simple subfunctors of indecomposable projective Mackey functors for a p-group P. Unlike the case of group algebras, the Mackey algebra is not in general self-injective. Thus, the socle of an indecomposable projective functor is not in general simple. We first show that the simple subfunctors of a projective functor P H , k , where H ⩽ P , are indexed by the normalizer in H of a subgroup of H. We then study the socle of a specific projective Mackey functor, namely the Burnside functor B P , and we focus on the case where P is abelian. In particular, our study enables us to determine the socle of an indecomposable projective Mackey functor indexed by a cyclic p-group, an abelian p-group of rank 2 and an elementary abelian p-group of rank 3.

Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules

The unified treatment of the five module-theoretic notions, transfer, inflation, transport of structure by an isomorphism, deflation and restriction, is given by the theory of biset functors, introduced by Bouc. In this paper, we construct the algebra realizing biset functors as representations. The algebra has a presentation similar to the well-known Mackey algebra. We adopt some natural constructions from the theory of Mackey functors and give two new constructions of simple biset functors. We also obtain a criterion for semisimplicity in terms of the biset functor version of the mark homomorphism. The criterion has an elementary generalization to arbitrary finite-dimensional algebras over a field.

A Swan length theorem and a Fong dimension theorem for Mackey algebras

We first present some results about Mackey algebras of p-groups over fields of characteristic p, including their primitive idempotents and decompositions of their simple and principal indecomposable modules under restriction. We then apply these results together with a Green's indecomposability theorem for Mackey algebras to obtain Mackey algebra versions of some classical results of group algebras which are mostly related to restriction, induction and dimensions of modules. Our results about dimensions include Mackey algebra analogues of Dickson's theorem, Swan's theorem and Fong's dimension formula.

Clifford theory for Mackey algebras

We develop a Clifford theory for Mackey algebras. For simple Mackey functors, using their classification we prove Mackey algebra versions of Clifford's theorem and the Clifford correspondence. Let μ R ( G ) be the Mackey algebra of a finite group G over a commutative unital ring R, and let 1 N be the unity of μ R ( N ) where N is a normal subgroup of G. Observing that 1 N μ R ( G ) 1 N is a crossed product of G / N over μ R ( N ) , a number of results concerning group graded algebras are extended to the context of Mackey algebras, including Fong's theorem, Green's indecomposibility theorem and some reduction and extension techniques for indecomposable Mackey functors.

The p-Blocks of the Mackey Algebra

Let p be a prime number. This paper describes the primitive idempotents and prime spectrum of the crossed Burnside algebra of a finite group over a p-local ring. The main application is a formula for the block idempotents of the p-local Mackey algebra of the group, in terms of the corresponding bloks of the group algebra.

On defect groups of the Mackey algebras for finite groups

On defect groups of the Mackey algebras for finite groups

The structure of Mackey functors

Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological K {\text {K}} -theory of classifying spaces, the algebraic K {\text {K}} -theory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the modules for a certain algebra, called the Mackey algebra. The study of Mackey functors is thus the same thing as the study of the representation theory of this algebra. We develop the properties of Mackey functors in the spirit of representation theory, and it emerges that there are great similarities with the representation theory of finite groups. In previous work we had classified the simple Mackey functors and demonstrated semisimplicity in characteristic zero. Here we consider the projective Mackey functors (in arbitrary characteristic), describing many of their features. We show, for example, that the Cartan matrix of the Mackey algebra may be computed from a decomposition matrix in the same way as for group representations. We determine the vertices, sources and Green correspondents of the projective and simple Mackey functors, as well as providing a way to compute the Ext groups for the simple Mackey functors. We parametrize the blocks of Mackey functors and determine the groups for which the Mackey algebra has finite representation type. It turns out that these Mackey algebras are direct sums of simple algebras and Brauer tree algebras. Throughout this theory there is a close connection between the properties of the Mackey functors, and the representations of the group on which they are defined, and of its subgroups. The relationships between these representations are exactly the information encoded by Mackey functors. This observation suggests the use of Mackey functors in a new way, as tools in group representation theory.

Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal p-sylow subgroup

Through recent investigations of J. Thrvenaz and P.J. Webb [4-8], the concept of Mackey functors, being originally introduced by J. A. Green [3] and studied by A.W.M. Dress, T. Yoshida, H. Sasaki, and others, has regained considerable interest. An important reason for this is ~i new interpretation of a conjecture of Aiperin in terms of Mackey functors in [8] using results of [7]. Given a commutative ring ,4, by its definition, it is immediate that the category of Mackey functors over .4 for a finite group G is an abelian category, even that it is the category of ,4-representations of a quiver with relations [8]. Namely a Mackey functor for G associates to each subgroup of G a ,(-module and moreover certain /'-linear maps between these spaces satisfying certain relations involving G. Thus a Mackey functor for G can be interpreted as a module over a path algebra of a quiver with relations, the so-called Mackey algebra [6, 9]. The main difference to the usual representation theory of quivers, and thus the main problem of interpreting Mackey functors as such representations, lies in the fact that the maps between these spaces do not lie in the radical of this path algebra, and this path algebra is not basic in general. So it is a natural problem to compute the quiver in the usual sense for the category of Mackey functors for a group G in particular in the case where ,4 is a field of characteristic p dividing the order of G. The question of when there are up to isomorphism only finitely many indecomposable Mackey functors for G led to recent investigations of Thrvenaz and Webb solving this problem in general, and it also led to the investigations we present in this paper which presents in a special situation, namely if G has a normal cyclic p-Sylow subgroup, an explicit computation of the quiver of the now so-called Mackcy algebra. By definition, the Mackey algebra is the basic ,(-algebra whose category of finitely generated