The structure of Mackey functors.

*(English)*Zbl 0834.20011Let \(G\) be a finite group and \(R\) a commutative ring. In most of the cases, \(R\) will be a field \(k\) of characteristic \(p > 0\), or a complete discrete valuation ring \(\mathcal O\) with quotient field \(K\) of characteristic 0 and residue field \(k\). By one of the definitions, a Mackey functor over \(R\) is an \(R\)-additive functor \(M : \Omega_R (G) \to R\)-mod, where the category \(\Omega_R (G)\) is constructed as follows. Firstly, let \(\omega_R(G)\) be the category of finite \(G\)- sets, where a morphism \(X \to Y\) is an equivalence class of diagrams of \(G\)-sets of the form \(X \leftarrow V \to Y\). The category \(\Omega_R (G)\) has the same objects as \(\omega_R (G)\), but \(\text{Hom}_{\Omega_R (G)} (X,Y)\) is the free \(R\)-module on the set \(\text{Hom}_{\omega_R (G)} (X,Y)\).

The authors show that the category of Mackey functors is equivalent to the category of modules of finite \(R\)-rank over the algebra \(\mu_R(G) = \bigoplus_{H,K \leq G} \text{Hom}_{\Omega_R (G)} (G/H, G/K)\), called the Mackey algebra. (This algebra may be also regarded as the path algebra of a quiver with relations.) Therefore, the study of Mackey functors can be done in the spirit of representation theory, and this is exactly the aim of this important paper. We proceed with a brief description of its content.

The composition factors of Mackey functors and the relationship between the Cartan matrix and the decomposition matrix of \(\mu_k (G)\) are investigated, and it is shown that this relation is the same as in the case of group representations. Consequently, the Cartan matrix of \(\mu_k (G)\) is symmetric and nonsingular.

The Burnside ring \(B(G)\) of \(G\) gives rise to a Mackey functor \(B^G\), called the Burnside Mackey functor. Certain fundamental projective Mackey functors are described in terms of the Burnside ring. Actually, \(B(G)\) acts as a ring of endomorphisms of any Mackey functor. This action can be interpreted in terms of the Mackey algebra, and as a consequence, a block theory for \(\mu_{\mathcal O} (G)\) is developed. More specifically, the identity element of \(B(G)\) is a sum of orthogonal idempotents \(f_J\) corresponding to the conjugacy classes of \(p\)-perfect subgroups \(J\) of \(G\) \(({\mathcal O}^p (J) = J)\). The category \(\text{Mack}_R(G, J)\) consisting of Mackey functors \(M\) for which \(f_J \cdot M = M\) is characterized in several ways, one of the tools being the vertex of a Mackey functor. Moreover, vertices, sources and Green correspondents of projective and of simple Mackey functors are completely determined when \(R = {\mathcal O}\).

If every prime divisor of \(|G |\) different from \(p\) is invertible in \(R\), and \(J\) is a \(p\)-perfect subgroup of \(G\), and \(\overline {N} = N_G(J) /J\), then the categories \(\text{Mack}_R (\overline {N},1)\) and \(\text{Mack}_R (G,K)\) are equivalent. This equivalence allows questions about Mackey functors to be reduced to the case of \(p\)-groups.

There is a bijection between indecomposable projective Mackey functors corresponding to \(p\)-subgroups, and indecomposable trivial source modules, given by the evaluation at the trivial subgroup 1; this, in turn, gives a strong connection with Hecke algebras.

The techniques developed in the paper allow to compute Ext groups for simple Mackey functors, and the lattice of subfunctors for certain Mackey functors, and also to describe the distribution of the simple Mackey functors into blocks.

The restriction and corestriction in group cohomology satisfy a certain relation which leads to the concept of cohomological Mackey functor. It turns out that these functors are modules for a quotient of \(\mu_R(G)\), called the cohomological Mackey algebra.

Some remarkable and surprising results are obtained in the last part of the paper. The \(k\)-algebra \(\mu_k(G)\) has finite representation type if and only if it is self injective, and this holds if and only if \(p^2\) does not divide the order of \(G\). In this case, each block of \(\mu_k (G)\) is either a Brauer tree algebra or a matrix algebra.

The similarity between the representation theory of \(\mu_R(G)\) and the representations of \(G\) and its subgroups is apparent at each stage, and the authors suggest that Mackey functors can be used as tools in group representation theory.

The authors show that the category of Mackey functors is equivalent to the category of modules of finite \(R\)-rank over the algebra \(\mu_R(G) = \bigoplus_{H,K \leq G} \text{Hom}_{\Omega_R (G)} (G/H, G/K)\), called the Mackey algebra. (This algebra may be also regarded as the path algebra of a quiver with relations.) Therefore, the study of Mackey functors can be done in the spirit of representation theory, and this is exactly the aim of this important paper. We proceed with a brief description of its content.

The composition factors of Mackey functors and the relationship between the Cartan matrix and the decomposition matrix of \(\mu_k (G)\) are investigated, and it is shown that this relation is the same as in the case of group representations. Consequently, the Cartan matrix of \(\mu_k (G)\) is symmetric and nonsingular.

The Burnside ring \(B(G)\) of \(G\) gives rise to a Mackey functor \(B^G\), called the Burnside Mackey functor. Certain fundamental projective Mackey functors are described in terms of the Burnside ring. Actually, \(B(G)\) acts as a ring of endomorphisms of any Mackey functor. This action can be interpreted in terms of the Mackey algebra, and as a consequence, a block theory for \(\mu_{\mathcal O} (G)\) is developed. More specifically, the identity element of \(B(G)\) is a sum of orthogonal idempotents \(f_J\) corresponding to the conjugacy classes of \(p\)-perfect subgroups \(J\) of \(G\) \(({\mathcal O}^p (J) = J)\). The category \(\text{Mack}_R(G, J)\) consisting of Mackey functors \(M\) for which \(f_J \cdot M = M\) is characterized in several ways, one of the tools being the vertex of a Mackey functor. Moreover, vertices, sources and Green correspondents of projective and of simple Mackey functors are completely determined when \(R = {\mathcal O}\).

If every prime divisor of \(|G |\) different from \(p\) is invertible in \(R\), and \(J\) is a \(p\)-perfect subgroup of \(G\), and \(\overline {N} = N_G(J) /J\), then the categories \(\text{Mack}_R (\overline {N},1)\) and \(\text{Mack}_R (G,K)\) are equivalent. This equivalence allows questions about Mackey functors to be reduced to the case of \(p\)-groups.

There is a bijection between indecomposable projective Mackey functors corresponding to \(p\)-subgroups, and indecomposable trivial source modules, given by the evaluation at the trivial subgroup 1; this, in turn, gives a strong connection with Hecke algebras.

The techniques developed in the paper allow to compute Ext groups for simple Mackey functors, and the lattice of subfunctors for certain Mackey functors, and also to describe the distribution of the simple Mackey functors into blocks.

The restriction and corestriction in group cohomology satisfy a certain relation which leads to the concept of cohomological Mackey functor. It turns out that these functors are modules for a quotient of \(\mu_R(G)\), called the cohomological Mackey algebra.

Some remarkable and surprising results are obtained in the last part of the paper. The \(k\)-algebra \(\mu_k(G)\) has finite representation type if and only if it is self injective, and this holds if and only if \(p^2\) does not divide the order of \(G\). In this case, each block of \(\mu_k (G)\) is either a Brauer tree algebra or a matrix algebra.

The similarity between the representation theory of \(\mu_R(G)\) and the representations of \(G\) and its subgroups is apparent at each stage, and the authors suggest that Mackey functors can be used as tools in group representation theory.

Reviewer: A.Marcus (Cluj-Napoca)

##### MSC:

20C20 | Modular representations and characters |

16G20 | Representations of quivers and partially ordered sets |

19A22 | Frobenius induction, Burnside and representation rings |

20J05 | Homological methods in group theory |

16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |