The structure of Mackey functors

Authors:
Jacques Thévenaz and Peter Webb

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1865-1961

MSC:
Primary 20C20; Secondary 20J05

DOI:
https://doi.org/10.1090/S0002-9947-1995-1261590-5

MathSciNet review:
1261590

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Abstract | References | Similar Articles | Additional Information

Abstract: 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 -theory of classifying spaces, the algebraic -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.

**[1]**J. L. Alperin,*Diagrams for modules*, J. Pure Appl. Algebra**16**(1980), no. 2, 111–119. MR**556154**, https://doi.org/10.1016/0022-4049(80)90010-9**[2]**J. L. Alperin,*Weights for finite groups*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 369–379. MR**933373****[3]**J. L. Alperin,*Local representation theory*, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR**860771****[4]**J. Alperin and Michel Broué,*Local methods in block theory*, Ann. of Math. (2)**110**(1979), no. 1, 143–157. MR**541333**, https://doi.org/10.2307/1971248**[5]**Michel Broué,*On Scott modules and 𝑝-permutation modules: an approach through the Brauer morphism*, Proc. Amer. Math. Soc.**93**(1985), no. 3, 401–408. MR**773988**, https://doi.org/10.1090/S0002-9939-1985-0773988-9**[6]**Charles W. Curtis and Irving Reiner,*Methods of representation theory. Vol. II*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR**892316****[7]**Tammo tom Dieck,*Transformation groups*, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR**889050****[8]**Larry Dornhoff,*Group representation theory. Part A: Ordinary representation theory*, Marcel Dekker, Inc., New York, 1971. Pure and Applied Mathematics, 7. MR**0347959**

Larry Dornhoff,*Group representation theory. Part B: Modular representation theory*, Marcel Dekker, Inc., New York, 1972. Pure and Applied Mathematics, 7. MR**0347960****[9]**Andreas W. M. Dress,*Contributions to the theory of induced representations*, Algebraic 𝐾-theory, II: “Classical” algebraic 𝐾-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 183–240. Lecture Notes in Math., Vol. 342. MR**0384917****[10]**P. Gabriel and Ch. Riedtmann,*Group representations without groups*, Comment. Math. Helv.**54**(1979), no. 2, 240–287. MR**535058**, https://doi.org/10.1007/BF02566271**[11]**J. A. Green,*Axiomatic representation theory for finite groups*, J. Pure Appl. Algebra**1**(1971), no. 1, 41–77. MR**0279208**, https://doi.org/10.1016/0022-4049(71)90011-9**[12]**D. G. Higman,*Indecomposable representations at characteristic 𝑝*, Duke Math. J.**21**(1954), 377–381. MR**0067896****[13]**P. J. Hilton and U. Stammbach,*A course in homological algebra*, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR**1438546****[14]**P. Landrock,*Finite group algebras and their modules*, London Mathematical Society Lecture Note Series, vol. 84, Cambridge University Press, Cambridge, 1983. MR**737910****[15]**Harald Lindner,*A remark on Mackey-functors*, Manuscripta Math.**18**(1976), no. 3, 273–278. MR**0401864**, https://doi.org/10.1007/BF01245921**[16]**Robert Oliver,*Whitehead groups of finite groups*, London Mathematical Society Lecture Note Series, vol. 132, Cambridge University Press, Cambridge, 1988. MR**933091****[17]**Hiroki Sasaki,*Green correspondence and transfer theorems of Wielandt type for 𝐺-functors*, J. Algebra**79**(1982), no. 1, 98–120. MR**679973**, https://doi.org/10.1016/0021-8693(82)90319-2**[18]**Daisuke Tambara,*Homological properties of the endomorphism rings of certain permutation modules*, Osaka J. Math.**26**(1989), no. 4, 807–828. MR**1040426****[19]**Jacques Thévenaz,*Some remarks on 𝐺-functors and the Brauer morphism*, J. Reine Angew. Math.**384**(1988), 24–56. MR**929977**, https://doi.org/10.1515/crll.1988.384.24**[20]**Jacques Thévenaz and Peter J. Webb,*Simple Mackey functors*, Proceedings of the Second International Group Theory Conference (Bressanone, 1989), 1990, pp. 299–319. MR**1068370****[21]**P. J. Webb,*A split exact sequence of Mackey functors*, Comment. Math. Helv.**66**(1991), no. 1, 34–69. MR**1090164**, https://doi.org/10.1007/BF02566635**[22]**Alfred Wiedemann,*Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal 𝑝-Sylow subgroup*, J. Algebra**150**(1992), no. 2, 296–307. MR**1176898**, https://doi.org/10.1016/S0021-8693(05)80033-X**[23]**Tomoyuki Yoshida,*Idempotents of Burnside rings and Dress induction theorem*, J. Algebra**80**(1983), no. 1, 90–105. MR**690705**, https://doi.org/10.1016/0021-8693(83)90019-4**[24]**Tomoyuki Yoshida,*On 𝐺-functors. II. Hecke operators and 𝐺-functors*, J. Math. Soc. Japan**35**(1983), no. 1, 179–190. MR**679083**, https://doi.org/10.2969/jmsj/03510179**[25]**Tomoyuki Yoshida,*Idempotents and transfer theorems of Burnside rings, character rings and span rings*, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 589–615. MR**1102277**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1261590-5

Keywords:
Mackey functor,
group cohomology,
Burnside ring,
group representation,
block,
Brauer tree

Article copyright:
© Copyright 1995
American Mathematical Society