The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups

Sambale, Benjamin

doi:10.1090/S0002-9939-2015-12637-8

The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups
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by Benjamin Sambale

Proc. Amer. Math. Soc. 143 (2015), 4291-4304

DOI: https://doi.org/10.1090/S0002-9939-2015-12637-8

Published electronically: April 1, 2015

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Abstract:

We prove the Alperin-McKay Conjecture for all $p$-blocks of finite groups with metacyclic, minimal non-abelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order $p$. In the special case $p=3$, we also verify Alperin’s Weight Conjecture for these defect groups. Moreover, in case $p=5$ we do the same for the non-abelian defect groups $C_{25}\rtimes C_{5^n}$. The proofs do not rely on the classification of the finite simple groups.

References

Jianbei An, Controlled blocks of the finite quasisimple groups for odd primes, Adv. Math. 227 (2011), no. 3, 1165–1194. MR 2799604, DOI 10.1016/j.aim.2011.03.003
Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011. MR 2848834, DOI 10.1017/CBO9781139003841
Richard Brauer, Defect groups in the theory of representations of finite groups, Illinois J. Math. 13 (1969), 53–73. MR 246979
Richard Brauer, On $2$-blocks with dihedral defect groups, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani & Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 367–393. MR 0354838
Michel Broué and Jørn B. Olsson, Subpair multiplicities in finite groups, J. Reine Angew. Math. 371 (1986), 125–143. MR 859323
E. C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 20–48. MR 200355, DOI 10.2307/1970529
Jill Dietz, Stable splittings of classifying spaces of metacyclic $p$-groups, $p$ odd, J. Pure Appl. Algebra 90 (1993), no. 2, 115–136. MR 1250764, DOI 10.1016/0022-4049(93)90125-D
Charles W. Eaton, Generalisations of conjectures of Brauer and Olsson, Arch. Math. (Basel) 81 (2003), no. 6, 621–626. MR 2029237, DOI 10.1007/s00013-003-0832-y
Charles W. Eaton, Radha Kessar, Burkhard Külshammer, and Benjamin Sambale, 2-blocks with abelian defect groups, Adv. Math. 254 (2014), 706–735. MR 3161112, DOI 10.1016/j.aim.2013.12.024
Charles W. Eaton and Alexander Moretó, Extending Brauer’s height zero conjecture to blocks with nonabelian defect groups, Int. Math. Res. Not. IMRN 20 (2014), 5581–5601. MR 3271182, DOI 10.1093/imrn/rnt131
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
Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
Mitsuo Fujii, On determinants of Cartan matrices of $p$-blocks, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 8, 401–403. MR 596014
Sheng Gao, On Brauer’s $k(B)$-problem for blocks with metacyclic defect groups of odd order, Arch. Math. (Basel) 96 (2011), no. 6, 507–512. MR 2821468, DOI 10.1007/s00013-011-0265-y
Sheng Gao, Blocks of full defect with nonabelian metacyclic defect groups, Arch. Math. (Basel) 98 (2012), no. 1, 1–12. MR 2885527, DOI 10.1007/s00013-011-0337-z
Sheng Gao and Ji-wen Zeng, On the number of ordinary irreducible characters in a $p$-block with a minimal nonabelian defect group, Comm. Algebra 39 (2011), no. 9, 3278–3297. MR 2845573, DOI 10.1080/00927872.2010.501775
Stuart Hendren, Extra special defect groups of order $p^3$ and exponent $p^2$, J. Algebra 291 (2005), no. 2, 457–491. MR 2163548, DOI 10.1016/j.jalgebra.2005.05.002
Lászlo Héthelyi, Burkhard Külshammer, and Benjamin Sambale, A note on Olsson’s conjecture, J. Algebra 398 (2014), 364–385. MR 3123771, DOI 10.1016/j.jalgebra.2012.08.010
Miles Holloway, Shigeo Koshitani, and Naoko Kunugi, Blocks with nonabelian defect groups which have cyclic subgroups of index $p$, Arch. Math. (Basel) 94 (2010), no. 2, 101–116. MR 2592757, DOI 10.1007/s00013-009-0075-7
H. Horimoto and A. Watanabe, On a perfect isometry between principal p-blocks of finite groups with cyclic p-hyperfocal subgroups, preprint.
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
I. M. Isaacs and Gabriel Navarro, New refinements of the McKay conjecture for arbitrary finite groups, Ann. of Math. (2) 156 (2002), no. 1, 333–344. MR 1935849, DOI 10.2307/3597192
R. Kessar, M. Linckelmann and G. Navarro, A characterisation of nilpotent blocks, arXiv:1402.5871v1.
Masao Kiyota, On $3$-blocks with an elementary abelian defect group of order $9$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 1, 33–58. MR 743518
Shigeo Koshitani and Naoko Kunugi, Broué’s conjecture holds for principal 3-blocks with elementary abelian defect group of order 9, J. Algebra 248 (2002), no. 2, 575–604. MR 1882112, DOI 10.1006/jabr.2001.9048
Shigeo Koshitani and Hyoue Miyachi, Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian Sylow 3-subgroup of order 9, Comm. Algebra 29 (2001), no. 10, 4509–4522. MR 1854148, DOI 10.1081/AGB-100106771
Burkhard Külshammer, On $p$-blocks of $p$-solvable groups, Comm. Algebra 9 (1981), no. 17, 1763–1785. MR 631888, DOI 10.1080/00927878108822682
Klaus Lux and Herbert Pahlings, Representations of groups, Cambridge Studies in Advanced Mathematics, vol. 124, Cambridge University Press, Cambridge, 2010. A computational approach. MR 2680716, DOI 10.1017/CBO9780511750915
Gunter Malle and Gabriel Navarro, Inequalities for some blocks of finite groups, Arch. Math. (Basel) 87 (2006), no. 5, 390–399. MR 2269921, DOI 10.1007/s00013-006-1769-8
Hirosi Nagao and Yukio Tsushima, Representations of finite groups, Academic Press, Inc., Boston, MA, 1989. Translated from the Japanese. MR 998775
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
Jørn B. Olsson, Lower defect groups, Comm. Algebra 8 (1980), no. 3, 261–288. MR 558114, DOI 10.1080/00927878008822458
Lluís Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), no. 1, 77–116. MR 943924, DOI 10.1007/BF01393688
Lluís Puig and Yoko Usami, Perfect isometries for blocks with abelian defect groups and Klein four inertial quotients, J. Algebra 160 (1993), no. 1, 192–225. MR 1237084, DOI 10.1006/jabr.1993.1184
Geoffrey R. Robinson, Large character heights, $Qd(p)$, and the ordinary weight conjecture, J. Algebra 319 (2008), no. 2, 657–679. MR 2381801, DOI 10.1016/j.jalgebra.2006.05.038
Geoffrey R. Robinson, On the focal defect group of a block, characters of height zero, and lower defect group multiplicities, J. Algebra 320 (2008), no. 6, 2624–2628. MR 2441776, DOI 10.1016/j.jalgebra.2008.04.032
Benjamin Sambale, Fusion systems on metacyclic 2-groups, Osaka J. Math. 49 (2012), no. 2, 325–329. MR 2945751
Benjamin Sambale, Brauer’s height zero conjecture for metacyclic defect groups, Pacific J. Math. 262 (2013), no. 2, 481–507. MR 3069071, DOI 10.2140/pjm.2013.262.481
Benjamin Sambale, Further evidence for conjectures in block theory, Algebra Number Theory 7 (2013), no. 9, 2241–2273. MR 3152013, DOI 10.2140/ant.2013.7.2241
Radu Stancu, Control of fusion in fusion systems, J. Algebra Appl. 5 (2006), no. 6, 817–837. MR 2286725, DOI 10.1142/S0219498806002034
Yoko Usami, On $p$-blocks with abelian defect groups and inertial index $2$ or $3$. I, J. Algebra 119 (1988), no. 1, 123–146. MR 971349, DOI 10.1016/0021-8693(88)90079-8
A. Watanabe, Appendix on blocks with elementary abelian defect group of order 9, in: Representation Theory of Finite Groups and Algebras, and Related Topics (Kyoto, 2008), 9–17, Kyoto University Research Institute for Mathematical Sciences, Kyoto, 2010.
Sheng Yang, On Olsson’s conjecture for blocks with metacyclic defect groups of odd order, Arch. Math. (Basel) 96 (2011), no. 5, 401–408. MR 2805343, DOI 10.1007/s00013-011-0251-4