Brauer’s generalized decomposition numbers and universal deformation rings

Bleher, Frauke

doi:10.1090/S0002-9947-2014-06120-5

Brauer’s generalized decomposition numbers and universal deformation rings
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by Frauke M. Bleher PDF

Trans. Amer. Math. Soc. 366 (2014), 6507-6540

Abstract:

The versal deformation ring $R(G,V)$ of a mod $p$ representation $V$ of a profinite group $G$ encodes all isomorphism classes of lifts of $V$ to representations of $G$ over complete local commutative Noetherian rings. We introduce a new technique for determining $R(G,V)$ when $G$ is finite which involves Brauer’s generalized decomposition numbers.

References

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, DOI 10.1017/CBO9780511623592
Frauke M. Bleher, Universal deformation rings and dihedral defect groups, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3661–3705. MR 2491895, DOI 10.1090/S0002-9947-09-04543-7
Frauke M. Bleher, Universal deformation rings and generalized quaternion defect groups, Adv. Math. 225 (2010), no. 3, 1499–1522. MR 2673738, DOI 10.1016/j.aim.2010.04.004
Frauke M. Bleher and Ted Chinburg, Universal deformation rings and cyclic blocks, Math. Ann. 318 (2000), no. 4, 805–836. MR 1802512, DOI 10.1007/s002080000148
Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 229–232 (English, with English and French summaries). MR 2196003, DOI 10.1016/j.crma.2005.12.006
Frauke M. Bleher and Ted Chinburg, Universal deformation rings need not be complete intersections, Math. Ann. 337 (2007), no. 4, 739–767. MR 2285736, DOI 10.1007/s00208-006-0054-2
Frauke M. Bleher, Ted Chinburg, and Bart de Smit, Inverse problems for deformation rings, Trans. Amer. Math. Soc. 365 (2013), no. 11, 6149–6165. MR 3091278, DOI 10.1090/S0002-9947-2013-05848-5
Frauke M. Bleher, Giovanna Llosent, and Jennifer B. Schaefer, Universal deformation rings and dihedral blocks with two simple modules, J. Algebra 345 (2011), 49–71. MR 2842053, DOI 10.1016/j.jalgebra.2011.08.010
V. M. Bondarenko and Ju. A. Drozd, The representation type of finite groups, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 71 (1977), 24–41, 282 (Russian). Modules and representations. MR 0472984
Richard Brauer, On the connection between the ordinary and the modular characters of groups of finite order, Ann. of Math. (2) 42 (1941), 926–935. MR 5732, DOI 10.2307/1968774
Richard Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung. II, Math. Z. 72 (1959/60), 25–46 (German). MR 108542, DOI 10.1007/BF01162934
Richard Brauer, Some applications of the theory of blocks of characters of finite groups. III, J. Algebra 3 (1966), 225–255. MR 202857, DOI 10.1016/0021-8693(66)90013-5
Richard Brauer, On blocks and sections in finite groups. II, Amer. J. Math. 90 (1968), 895–925. MR 244398, DOI 10.2307/2373490
Richard Brauer, Some applications of the theory of blocks of characters of finite groups. IV, J. Algebra 17 (1971), 489–521. MR 281806, DOI 10.1016/0021-8693(71)90006-8
Richard Brauer, On $2$-blocks with dihedral defect groups, Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 367–393. MR 0354838
Sheila Brenner, Modular representations of $p$ groups, J. Algebra 15 (1970), 89–102. MR 255704, DOI 10.1016/0021-8693(70)90088-8
M. C. R. Butler and Claus Michael Ringel, Auslander-Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), no. 1-2, 145–179. MR 876976, DOI 10.1080/00927878708823416
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
Ted Chinburg, Can deformation rings of group representations not be local complete intersections? In: Problems from the Workshop on Automorphisms of Curves. Edited by Gunther Cornelissen and Frans Oort, with contributions by I. Bouw, T. Chinburg, Cornelissen, C. Gasbarri, D. Glass, C. Lehr, M. Matignon, Oort, R. Pries and S. Wewers. Rend. Sem. Mat. Univ. Padova 113 (2005), 129–177.
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
E. C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 20–48. MR 200355, DOI 10.2307/1970529
Bart de Smit and Hendrik W. Lenstra Jr., Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 313–326. MR 1638482
Florian Eisele, $p$-adic lifting problems and derived equivalences, J. Algebra 356 (2012), 90–114. MR 2891124, DOI 10.1016/j.jalgebra.2012.01.015
Karin Erdmann, Blocks whose defect groups are Klein four groups: a correction, J. Algebra 76 (1982), no. 2, 505–518. MR 661869, DOI 10.1016/0021-8693(82)90228-9
K. Erdmann, Algebras and semidihedral defect groups. I, Proc. London Math. Soc. (3) 57 (1988), no. 1, 109–150. MR 940432, DOI 10.1112/plms/s3-57.1.109
Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
Paul Fong, A note on splitting fields of representations of finite groups, Illinois J. Math. 7 (1963), 515–520. MR 153741
D. G. Higman, Indecomposable representations at characteristic $p$, Duke Math. J. 21 (1954), 377–381. MR 67896, DOI 10.1215/S0012-7094-54-02138-9
Thorsten Holm, Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Algebra 211 (1999), no. 1, 159–205. MR 1656577, DOI 10.1006/jabr.1998.7544
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
Henning Krause, Maps between tree and band modules, J. Algebra 137 (1991), no. 1, 186–194. MR 1090218, DOI 10.1016/0021-8693(91)90088-P
Markus Linckelmann, A derived equivalence for blocks with dihedral defect groups, J. Algebra 164 (1994), no. 1, 244–255. MR 1268334, DOI 10.1006/jabr.1994.1061
Markus Linckelmann, The source algebras of blocks with a Klein four defect group, J. Algebra 167 (1994), no. 3, 821–854. MR 1287072, DOI 10.1006/jabr.1994.1214
B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 10.1007/978-1-4613-9649-9_{7}
Jørn Børling Olsson, On $2$-blocks with quaternion and quasidihedral defect groups, J. Algebra 36 (1975), no. 2, 212–241. MR 376841, DOI 10.1016/0021-8693(75)90099-X