The structure of Mackey functors
Authors:
Jacques Thévenaz and Peter Webb
Journal:
Trans. Amer. Math. Soc. 347 (1995), 18651961
MSC:
Primary 20C20; Secondary 20J05
MathSciNet review:
1261590
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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
(81h:16047), http://dx.doi.org/10.1016/00224049(80)900109
 [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
(89h:20015)
 [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
(87i:20002)
 [4]
J.
Alperin and Michel
Broué, Local methods in block theory, Ann. of Math. (2)
110 (1979), no. 1, 143–157. MR 541333
(80f:20010), http://dx.doi.org/10.2307/1971248
 [5]
Michel
Broué, On Scott modules and
𝑝permutation modules: an approach through the Brauer
morphism, Proc. Amer. Math. Soc.
93 (1985), no. 3,
401–408. MR
773988 (86d:20010), http://dx.doi.org/10.1090/S00029939198507739889
 [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 WileyInterscience
Publication. MR
892316 (88f:20002)
 [7]
Tammo
tom Dieck, Transformation groups, de Gruyter Studies in
Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050
(89c:57048)
 [8]
Larry
Dornhoff, Group representation theory. Part B: Modular
representation theory, Marcel Dekker, Inc., New York, 1972. Pure and
Applied Mathematics, 7. MR 0347960
(50 #458b)
 [9]
Andreas
W. M. Dress, Contributions to the theory of induced
representations, Algebraic 𝐾theory, II:
“Classical” algebraic 𝐾theory 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 (52 #5787)
 [10]
P.
Gabriel and Ch.
Riedtmann, Group representations without groups, Comment.
Math. Helv. 54 (1979), no. 2, 240–287. MR 535058
(80k:16040), http://dx.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 (43 #4931)
 [12]
D.
G. Higman, Indecomposable representations at characteristic
𝑝, Duke Math. J. 21 (1954), 377–381. MR 0067896
(16,794c)
 [13]
P.
J. Hilton and U.
Stammbach, A course in homological algebra, 2nd ed., Graduate
Texts in Mathematics, vol. 4, SpringerVerlag, New York, 1997. MR 1438546
(97k:18001)
 [14]
P.
Landrock, Finite group algebras and their modules, London
Mathematical Society Lecture Note Series, vol. 84, Cambridge
University Press, Cambridge, 1983. MR 737910
(85h:20002)
 [15]
Harald
Lindner, A remark on Mackeyfunctors, Manuscripta Math.
18 (1976), no. 3, 273–278. MR 0401864
(53 #5691)
 [16]
Robert
Oliver, Whitehead groups of finite groups, London Mathematical
Society Lecture Note Series, vol. 132, Cambridge University Press,
Cambridge, 1988. MR 933091
(89h:18014)
 [17]
Hiroki
Sasaki, Green correspondence and transfer theorems of Wielandt type
for 𝐺functors, J. Algebra 79 (1982),
no. 1, 98–120. MR 679973
(84a:20018), http://dx.doi.org/10.1016/00218693(82)903192
 [18]
Daisuke
Tambara, Homological properties of the endomorphism rings of
certain permutation modules, Osaka J. Math. 26
(1989), no. 4, 807–828. MR 1040426
(91a:20060)
 [19]
Jacques
Thévenaz, Some remarks on 𝐺functors and the Brauer
morphism, J. Reine Angew. Math. 384 (1988),
24–56. MR
929977 (89b:20035), http://dx.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
(91g:20011)
 [21]
P.
J. Webb, A split exact sequence of Mackey functors, Comment.
Math. Helv. 66 (1991), no. 1, 34–69. MR 1090164
(92c:20095), http://dx.doi.org/10.1007/BF02566635
 [22]
Alfred
Wiedemann, Elementary construction of the quiver of the Mackey
algebra for groups with cyclic normal 𝑝Sylow subgroup, J.
Algebra 150 (1992), no. 2, 296–307. MR 1176898
(93g:20024), http://dx.doi.org/10.1016/S00218693(05)80033X
 [23]
Tomoyuki
Yoshida, Idempotents of Burnside rings and Dress induction
theorem, J. Algebra 80 (1983), no. 1,
90–105. MR
690705 (85d:20004), http://dx.doi.org/10.1016/00218693(83)900194
 [24]
Tomoyuki
Yoshida, On 𝐺functors. II. Hecke operators and
𝐺functors, J. Math. Soc. Japan 35 (1983),
no. 1, 179–190. MR 679083
(84b:20010), http://dx.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
 [1]
 J.L. Alperin, Diagrams for modules, J. Pure Appl. Algebra 16 (1980), 111119. MR 556154 (81h:16047)
 [2]
 , Weights for finite groups, The Arcata Conference on Representations of Finite Groups (P. Fong, ed.), Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 369379. MR 933373 (89h:20015)
 [3]
 , Local representation theory, Cambridge University Press, 1986. MR 860771 (87i:20002)
 [4]
 J.L. Alperin and M. Broué, Local methods in block theory, Ann. of Math. 110 (1979), 143157. MR 541333 (80f:20010)
 [5]
 M. Broué, On Scott modules and permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), 401408. MR 773988 (86d:20010)
 [6]
 C.W. Curtis and I. Reiner, Methods of representation theory II, Wiley, 1987. MR 892316 (88f:20002)
 [7]
 T. tom Dieck, Transformation groups, De Gruyter, BerlinNew York, 1987. MR 889050 (89c:57048)
 [8]
 L. Dornhoff, Group representation theory, Marcel Dekker, New York, 1972. MR 0347960 (50:458b)
 [9]
 A.W.M. Dress, Contributions to the theory of induced representations, Algebraic Theory II (H. Bass, ed.), Lecture Notes in Math., vol. 342, SpringerVerlag, 1973, pp. 183240. MR 0384917 (52:5787)
 [10]
 P. Gabriel and Ch. Riedtmann, Group representations without groups, Comment. Math. Helv. 54 (1979), 240287. MR 535058 (80k:16040)
 [11]
 J.A. Green, Axiomatic representation theory for finite groups, J. Pure Appl. Algebra 1 (1971), 4177. MR 0279208 (43:4931)
 [12]
 D.G. Higman, Indecomposable representations at characteristic , Duke Math. J. 21 (1954), 369376. MR 0067896 (16:794c)
 [13]
 P.J. Hilton and U. Stammbach, A course in homological algebra, Graduate Texts in Math. , SpringerVerlag, 1970. MR 1438546 (97k:18001)
 [14]
 P. Landrock, Finite group algebras and their modules, LMS Lecture Notes 84, Cambridge University Press, 1983. MR 737910 (85h:20002)
 [15]
 H. Lindner, A remark on Mackey functors, Manuscripta Math. 18 (1976), 273278. MR 0401864 (53:5691)
 [16]
 R. Oliver, Whitehead groups of finite groups, LMS Lecture Notes 132, Cambridge University Press, 1988. MR 933091 (89h:18014)
 [17]
 H. Sasaki, Green correspondence and transfer theorems of Wielandt type for functors, J. Algebra 79 (1982), 98120. MR 679973 (84a:20018)
 [18]
 D. Tambara, Homological properties of the endomorphism ring of certain permutation modules, Osaka J. Math. 26 (1989), 807828. MR 1040426 (91a:20060)
 [19]
 J. Thévenaz, Some remarks on functors and the Brauer morphism, J. Reine Angew. Math. 384 (1988), 2456. MR 929977 (89b:20035)
 [20]
 J. Thévenaz and P.J. Webb, Simple Mackey functors, Proc. of 2nd International Group Theory Conference, Bressanone (1989), Supplemento ai Rendiconti del Circolo Matematico di Palermo 23, 1990, pp. 299319. MR 1068370 (91g:20011)
 [21]
 P.J. Webb, A split exact sequence of Mackey functors, Comment. Math. Helv. 66 (1991), 3469. MR 1090164 (92c:20095)
 [22]
 A. Wiedemann, Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal Sylow subgroup, J. Algebra 150 (1992), 296307. MR 1176898 (93g:20024)
 [23]
 T. Yoshida, Idempotents of Burnside rings and Dress induction theorem, J. Algebra 80 (1983), 90105. MR 690705 (85d:20004)
 [24]
 , On functors II: Hecke operators and functors, J. Math. Soc. Japan 35 (1983), 179190. MR 679083 (84b:20010)
 [25]
 , Idempotents and transfer theorems of Burnside rings, character rings and span rings, Algebraic and Topological Theories (to the memory of T. Miyata) (1985), 589615. MR 1102277
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512615905
PII:
S 00029947(1995)12615905
Keywords:
Mackey functor,
group cohomology,
Burnside ring,
group representation,
block,
Brauer tree
Article copyright:
© Copyright 1995
American Mathematical Society
