Categorical constructions related to finite groups

Oda, Fumihito; Nakaoka, Hiroyuki

doi:10.1090/suga/438

Categorical constructions related to finite groups
HTML articles powered by AMS MathViewer

by Fumihito Oda and Hiroyuki Nakaoka
Translated by: Fumihito Oda and Hiroyuki Nakaoka

Sugaku Expositions 32 (2019), 57-86

DOI: https://doi.org/10.1090/suga/438

Published electronically: March 21, 2019

PDF | Request permission

Abstract:

In this survey, we introduce categorical notions related to finite groups, which have been developed to describe the behavior of algebraic systems appearing in the finite group theory. Particularly, we focus on Mackey, Green, and biset functors, in relation with Dress construction, transfer theorem, Alperin’s weight conjecture, and Dade groups.

References

Alperin, J. L.: Invertible modules for groups, Notices Amer. Math. Soc., 24 (1977).
Alperin, J. L.: Weights for Finite groups, in: ed. P. Fong, The Arcata Conference on Representations of Finite Groups, (1987) 369–379, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI.
J. L. Alperin, A construction of endo-permutation modules, J. Group Theory 4 (2001), no. 1, 3–10. MR 1808834, DOI 10.1515/jgth.2001.005
D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
Boltje, R.: Mackey functors and related structures in representation theory and number theory, Habilitation-Thesis, Universität Augsburg (1995).
Robert Boltje and Susanne Danz, A ghost ring for the left-free double Burnside ring and an application to fusion systems, Adv. Math. 229 (2012), no. 3, 1688–1733. MR 2871154, DOI 10.1016/j.aim.2011.12.015
Serge Bouc, Green functors and $G$-sets, Lecture Notes in Mathematics, vol. 1671, Springer-Verlag, Berlin, 1997. MR 1483069, DOI 10.1007/BFb0095821
S. Bouc, Tensor induction of relative syzygies, J. Reine Angew. Math. 523 (2000), 113–171. MR 1762958, DOI 10.1515/crll.2000.045
Serge Bouc, Burnside rings, Handbook of algebra, Vol. 2, Handb. Algebr., vol. 2, Elsevier/North-Holland, Amsterdam, 2000, pp. 739–804. MR 1759611, DOI 10.1016/S1570-7954(00)80043-1
Serge Bouc, A remark on a theorem of Ritter and Segal, J. Group Theory 4 (2001), no. 1, 11–18. MR 1808835, DOI 10.1515/jgth.2001.002
Serge Bouc, Hochschild constructions for Green functors, Comm. Algebra 31 (2003), no. 1, 403–436. MR 1969231, DOI 10.1081/AGB-120016767
Serge Bouc, The $p$-blocks of the Mackey algebra, Algebr. Represent. Theory 6 (2003), no. 5, 515–543. MR 2026725, DOI 10.1023/B:ALGE.0000006493.77655.e5
Serge Bouc, The functor of rational representations for $p$-groups, Adv. Math. 186 (2004), no. 2, 267–306. MR 2073907, DOI 10.1016/j.aim.2003.08.002
Serge Bouc, A remark on the Dade group and the Burnside group, J. Algebra 279 (2004), no. 1, 180–190. MR 2078393, DOI 10.1016/j.jalgebra.2003.10.003
Serge Bouc, Biset functors and genetic sections for $p$-groups, J. Algebra 284 (2005), no. 1, 179–202. MR 2115011, DOI 10.1016/j.jalgebra.2004.06.034
Serge Bouc, The Dade group of a $p$-group, Invent. Math. 164 (2006), no. 1, 189–231. MR 2207787, DOI 10.1007/s00222-005-0476-6
Serge Bouc, The functor of units of Burnside rings for $p$-groups, Comment. Math. Helv. 82 (2007), no. 3, 583–615. MR 2314054, DOI 10.4171/CMH/103
Serge Bouc, Biset functors for finite groups, Lecture Notes in Mathematics, vol. 1990, Springer-Verlag, Berlin, 2010. MR 2598185, DOI 10.1007/978-3-642-11297-3
Serge Bouc and Nadia Mazza, The Dade group of (almost) extraspecial $p$-groups, J. Pure Appl. Algebra 192 (2004), no. 1-3, 21–51. MR 2067187, DOI 10.1016/j.jpaa.2004.02.008
Serge Bouc and Jacques Thévenaz, The group of endo-permutation modules, Invent. Math. 139 (2000), no. 2, 275–349. MR 1738450, DOI 10.1007/s002229900026
Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856. MR 1992826, DOI 10.1090/S0894-0347-03-00434-X
Kenneth S. Brown, Euler characteristics of groups: the $p$-fractional part, Invent. Math. 29 (1975), no. 1, 1–5. MR 385008, DOI 10.1007/BF01405170
Morten Brun, Witt vectors and Tambara functors, Adv. Math. 193 (2005), no. 2, 233–256. MR 2136887, DOI 10.1016/j.aim.2004.05.002
W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 0069818
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
Jon F. Carlson, A characterization of endotrivial modules over $p$-groups, Manuscripta Math. 97 (1998), no. 3, 303–307. MR 1654772, DOI 10.1007/s002290050102
Jon F. Carlson and Jacques Thévenaz, Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000), no. 4, 303–335. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. MR 1808129, DOI 10.1023/A:1009988424910
Jon F. Carlson and Jacques Thévenaz, The classification of endo-trivial modules, Invent. Math. 158 (2004), no. 2, 389–411. MR 2096798, DOI 10.1007/s00222-004-0385-0
Jon F. Carlson and Jacques Thévenaz, The classification of torsion endo-trivial modules, Ann. of Math. (2) 162 (2005), no. 2, 823–883. MR 2183283, DOI 10.4007/annals.2005.162.823
Everett C. Dade, Endo-permutation modules over $p$-groups. I, Ann. of Math. (2) 107 (1978), no. 3, 459–494. MR 496843, DOI 10.2307/1971125
Everett Dade, Endo-permutation modules over $p$-groups. II, Ann. of Math. (2) 108 (1978), no. 2, 317–346. MR 506990, DOI 10.2307/1971169
Antonio Díaz and Assaf Libman, The Burnside ring of fusion systems, Adv. Math. 222 (2009), no. 6, 1943–1963. MR 2562769, DOI 10.1016/j.aim.2009.06.023
Andreas Dress, A characterisation of solvable groups, Math. Z. 110 (1969), 213–217. MR 248239, DOI 10.1007/BF01110213
Andreas Dress, Operations in representation rings, Representation theory of finite groups and related topics (Proc. Sympos. Pure Math., Vol. XXI, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 39–45. MR 0325740
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) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 183–240. MR 0384917
Andreas W. M. Dress and Christian Siebeneicher, The Burnside ring of profinite groups and the Witt vector construction, Adv. in Math. 70 (1988), no. 1, 87–132. MR 947758, DOI 10.1016/0001-8708(88)90052-7
Andreas W. M. Dress and Christian Siebeneicher, The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, $\lambda$-rings, and the universal ring of Witt vectors, Adv. Math. 78 (1989), no. 1, 1–41. MR 1021547, DOI 10.1016/0001-8708(89)90027-3
Andreas W. M. Dress, Christian Siebeneicher, and Tomoyuki Yoshida, An application of Burnside rings in elementary finite group theory, Adv. Math. 91 (1992), no. 1, 27–44. MR 1144342, DOI 10.1016/0001-8708(92)90008-9
Jesse Elliott, Constructing Witt-Burnside rings, Adv. Math. 203 (2006), no. 1, 319–363. MR 2231048, DOI 10.1016/j.aim.2005.04.014
Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. MR 166261
Nicola Gambino and Joachim Kock, Polynomial functors and polynomial monads, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 153–192. MR 3002590, DOI 10.1017/S0305004112000394
David Gluck, Idempotent formula for the Burnside algebra with applications to the $p$-subgroup simplicial complex, Illinois J. Math. 25 (1981), no. 1, 63–67. MR 602896
Daniel Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co., New York, 1980. MR 569209
J. A. Green, Axiomatic representation theory for finite groups, J. Pure Appl. Algebra 1 (1971), no. 1, 41–77. MR 279208, DOI 10.1016/0022-4049(71)90011-9
E. T. Jacobson, The Brauer ring of a field, Illinois J. Math. 30 (1986), no. 3, 479–510. MR 850345
Harald Lindner, A remark on Mackey-functors, Manuscripta Math. 18 (1976), no. 3, 273–278. MR 401864, DOI 10.1007/BF01245921
Toshimitsu Matsuda, On the unit groups of Burnside rings, Japan. J. Math. (N.S.) 8 (1982), no. 1, 71–93. MR 722522, DOI 10.4099/math1924.8.71
Toshimitsu Matsuda and Takehiko Miyata, On the unit groups of Burnside rings of finite groups, J. Math. Soc. Japan 35 (1983), no. 2, 345–354. MR 692332, DOI 10.2969/jmsj/03520345
N. Metropolis and Gian-Carlo Rota, Witt vectors and the algebra of necklaces, Adv. in Math. 50 (1983), no. 2, 95–125. MR 723197, DOI 10.1016/0001-8708(83)90035-X
Hiroyuki Nakaoka, Tambara functors on profinite groups and generalized Burnside functors, Comm. Algebra 37 (2009), no. 9, 3095–3151. MR 2554193, DOI 10.1080/00927870902747605
Hiroyuki Nakaoka, Tambarization of a Mackey functor and its application to the Witt-Burnside construction, Adv. Math. 227 (2011), no. 5, 2107–2143. MR 2803797, DOI 10.1016/j.aim.2011.04.015
Hiroyuki Nakaoka, On the fractions of semi-Mackey and Tambara functors, J. Algebra 352 (2012), 79–103. MR 2862176, DOI 10.1016/j.jalgebra.2011.11.013
Hiroyuki Nakaoka, Ideals of Tambara functors, Adv. Math. 230 (2012), no. 4-6, 2295–2331. MR 2927371, DOI 10.1016/j.aim.2012.04.021
Nakaoka, H.: Some algebraic operations on Tambara functors on finite groups. Cohomology Theory of Finite Groups and Related Topics. (Japanese) Sūrikaisekikenkyūsho Kōkyūroku 1784 (2012) 161–174.
Hiroyuki Nakaoka, A generalization of the Dress construction for a Tambara functor, and its relation to polynomial Tambara functors, Adv. Math. 235 (2013), 237–260. MR 3010058, DOI 10.1016/j.aim.2012.11.013
Hiroyuki Nakaoka, The spectrum of the Burnside Tambara functor on a finite cyclic $p$-group, J. Algebra 398 (2014), 21–54. MR 3123752, DOI 10.1016/j.jalgebra.2013.09.010
Tetsuro Okuyama, Hiroki Sasaki, and Akihiko Hida, Cohomology theory for finite groups, Sūgaku 62 (2010), no. 2, 240–266 (Japanese). MR 2641973
Fumihito Oda, Crossed Burnside rings and Bouc’s construction of Green functors, J. Algebra 315 (2007), no. 1, 18–30. MR 2344331, DOI 10.1016/j.jalgebra.2007.04.024
Fumihito Oda, The generalized Burnside ring with respect to $p$-centric subgroups, J. Algebra 320 (2008), no. 10, 3726–3732. MR 2457719, DOI 10.1016/j.jalgebra.2008.08.004
Fumihito Oda, The crossed Burnside ring, the Drinfel’d double, and the Dade group of a $p$-group, Algebr. Represent. Theory 13 (2010), no. 2, 231–242. MR 2601543, DOI 10.1007/s10468-008-9117-2
Oda, F.: The Tambara functors and the crossed Burnside rings. Cohomology Theory of Finite Groups and Related Topics. (Japanese) Sūrikaisekikenkyūsho Kōkyūroku 1784 (2012) 155–160.
Fumihito Oda and Masato Sawabe, A collection of subgroups for the generalized Burnside ring, Adv. Math. 222 (2009), no. 1, 307–317. MR 2531376, DOI 10.1016/j.aim.2009.04.005
Fumihito Oda and Masato Sawabe, The generalized Burnside rings with respect to a collection of self-normalizing subgroups, J. Algebra 334 (2011), 219–231. MR 2787660, DOI 10.1016/j.jalgebra.2011.02.004
Fumihito Oda and Tomoyuki Yoshida, Crossed Burnside rings. I. The fundamental theorem, J. Algebra 236 (2001), no. 1, 29–79. MR 1808345, DOI 10.1006/jabr.2000.8341
Fumihito Oda and Tomoyuki Yoshida, On the generalized Burnside ring with respect to the Young subgroups of the symmetric group, J. Algebra 236 (2001), no. 1, 349–354. MR 1808357, DOI 10.1006/jabr.2000.8510
Fumihito Oda and Tomoyuki Yoshida, Crossed Burnside rings. II. The Dress construction of a Green functor, J. Algebra 282 (2004), no. 1, 58–82. MR 2095572, DOI 10.1016/j.jalgebra.2004.07.040
Fumihito Oda and Tomoyuki Yoshida, The crossed Burnside rings III: the dress construction for a Tambara functor, J. Algebra 327 (2011), 31–49. MR 2746028, DOI 10.1016/j.jalgebra.2010.10.018
Elango Panchadcharam and Ross Street, Mackey functors on compact closed categories, J. Homotopy Relat. Struct. 2 (2007), no. 2, 261–293. MR 2369169
Lluís Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), no. 1, 77–116. MR 943924, DOI 10.1007/BF01393688
Lluís Puig, Affirmative answer to a question of Feit, J. Algebra 131 (1990), no. 2, 513–526. MR 1058563, DOI 10.1016/0021-8693(90)90192-Q
Lluís Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks, Progress in Mathematics, vol. 178, Birkhäuser Verlag, Basel, 1999. MR 1707300
Peter Roquette, Realisierung von Darstellungen endlicher nilpotenter Gruppen, Arch. Math. (Basel) 9 (1958), 241–250 (German). MR 97452, DOI 10.1007/BF01900587
Victor P. Snaith, Explicit Brauer induction, Cambridge Studies in Advanced Mathematics, vol. 40, Cambridge University Press, Cambridge, 1994. With applications to algebra and number theory. MR 1310780, DOI 10.1017/CBO9780511600746
Louis Solomon, The Burnside algebra of a finite group, J. Combinatorial Theory 2 (1967), 603–615. MR 214679
Strickland, N.: Tambara functors, arXiv:1205.2516.
Yugen Takegahara, Multiple Burnside rings and Brauer induction formulae, J. Algebra 324 (2010), no. 7, 1656–1686. MR 2673756, DOI 10.1016/j.jalgebra.2010.05.021
Takegahara, Y.: Generalizations of Burnside ring and their applications. Research into Vertex Operator Algebras, Finite Groups and Combinatorics. Sūrikaisekikenkyūsho Kōkyūroku 1756 (2011) 41–50.
Takegahara, Y.: A generalization of Burnside rings and its application. Research into Vertex Operator Algebras, Finite Groups and Combinatorics. (Japanese) Sūrikaisekikenkyūsho Kōkyūroku 1811 (2012) 60–77.
Tambara, D.: Multiplicative transfer and Mackey functor, unpublished.
D. Tambara, On multiplicative transfer, Comm. Algebra 21 (1993), no. 4, 1393–1420. MR 1209937, DOI 10.1080/00927879308824627
D. Tambara, A partial Burnside ring of $\textrm {GL}(n,q)$ relative to line stabilizers, J. Algebra 296 (2006), no. 1, 301–322. MR 2192609, DOI 10.1016/j.jalgebra.2005.06.007
Jacques Thévenaz, Some remarks on $G$-functors and the Brauer morphism, J. Reine Angew. Math. 384 (1988), 24–56. MR 929977, DOI 10.1515/crll.1988.384.24
Jacques Thévenaz, A visit to the kingdom of the Mackey functors, Bayreuth. Math. Schr. 33 (1990), 215–241. Darstellungstheorietage (Sion, 1989). MR 1063878
Jacques Thévenaz, Locally determined functions and Alperin’s conjecture, J. London Math. Soc. (2) 45 (1992), no. 3, 446–468. MR 1180255, DOI 10.1112/jlms/s2-45.3.446
Jacques Thévenaz, Equivariant $K$-theory and Alperin’s conjecture, J. Pure Appl. Algebra 85 (1993), no. 2, 185–202. MR 1207508, DOI 10.1016/0022-4049(93)90052-U
Jacques Thévenaz, Endo-permutation modules, a guided tour, Group representation theory, EPFL Press, Lausanne, 2007, pp. 115–147. MR 2336639
J. Thévenaz and P. J. Webb, A Mackey functor version of a conjecture of Alperin, Astérisque 181-182 (1990), 9, 263–272 (English, with French summary). MR 1051255
Jacques Thévenaz and Peter Webb, The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), no. 6, 1865–1961. MR 1261590, DOI 10.1090/S0002-9947-1995-1261590-5
Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. MR 551743
Katsuhiro Uno and Naoko Kunugi, Conjectures in the modular representation theory of finite groups, Sūgaku 65 (2013), no. 1, 1–23 (Japanese). MR 3057843
Ergün Yalçın, An induction theorem for the unit groups of Burnside rings of 2-groups, J. Algebra 289 (2005), no. 1, 105–127. MR 2139093, DOI 10.1016/j.jalgebra.2005.03.029
Tomoyuki Yoshida, Transfer theory in topoi. Use of finite group theory, Sūgaku 32 (1980), no. 3, 193–212 (Japanese). MR 613776
Tomoyuki Yoshida, Idempotents of Burnside rings and Dress induction theorem, J. Algebra 80 (1983), no. 1, 90–105. MR 690705, DOI 10.1016/0021-8693(83)90019-4
Tomoyuki Yoshida, On the unit groups of Burnside rings, J. Math. Soc. Japan 42 (1990), no. 1, 31–64. MR 1027539, DOI 10.2969/jmsj/04210031
Tomoyuki Yoshida, The generalized Burnside ring of a finite group, Hokkaido Math. J. 19 (1990), no. 3, 509–574. MR 1078504, DOI 10.14492/hokmj/1381517496
Tomoyuki Yoshida, Classical problems of group theory. I. Counting subgroups and homomorphisms, Sūgaku 45 (1993), no. 3, 193–207 (Japanese, with Japanese summary). MR 1256464
Tomoyuki Yoshida, Crossed $G$-sets and crossed Burnside rings, Sūrikaisekikenkyūsho K\B{o}kyūroku 991 (1997), 1–15. Group theory and combinatorial mathematics (Japanese) (Kyoto, 1996). MR 1608373
Tomoyuki Yoshida, Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories, J. Algebra 240 (2001), no. 1, 40–82. MR 1830543, DOI 10.1006/jabr.2000.8602
Yoshida, T.: Polynomial Rings with coefficients in Tambara Functors. Cohomology Theory of Finite Groups and Related Topics. (Japanese) Sūrikaisekikenkyūsho Kōkyūroku 1466 (2006) 21–34.
Peter Webb, A guide to Mackey functors, Handbook of algebra, Vol. 2, Handb. Algebr., vol. 2, Elsevier/North-Holland, Amsterdam, 2000, pp. 805–836. MR 1759612, DOI 10.1016/S1570-7954(00)80044-3
S. J. Witherspoon, The representation ring of the quantum double of a finite group, J. Algebra 179 (1996), no. 1, 305–329. MR 1367852, DOI 10.1006/jabr.1996.0014