Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64

C. G. Ardito

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 220-239

Résumé

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Détail

Keywords: Donovan's conjecture; finite groups; Morita equivalence; block theory; modular representation theory.

Bibliographie

[1] Alperin J.L., Local representation theory, Ser. Cambridge Studies in Advanced mathematics, 11, Cambridge University Press, Cambridge, 1986, 178 pp. | MR | Zbl

[2] Ardito C.G., Morita equivalence classes of blocks with elementary abelian defect groups of order 32, 2019, 32 pp., arXiv: math.1908.02652 | MR

[3] Ardito C.G., Blocks with an elementary abelian defect group in characteristic two, PhD Thesis, University of Manchester, 2020, 256 pp. | MR

[4] Boltje R., Kessar R., Linckelmann M., “On Picard groups of blocks of finite groups”, J. Algebra, 558 (2020), 70–101 | DOI | MR | Zbl

[5] Bosma W., Cannon J.J., Fieker C., Steel A., Handbook of Magma functions, 2.16, 2010

[6] Craven D., Eaton C.W., Kessar R. and Linckelmann M., “The structure of blocks with a Klein four defect group”, Math. Z., 268:1 (2011), 441–476 | DOI | MR | Zbl

[7] Craven D., Rouquier R., “Perverse equivalences and Broue's conjecture”, Advances in Mathematics, 248 (2013), 1–58 | DOI | MR | Zbl

[8] Eaton C.W., Kessar R., Kulshammer B.K and Sambale B., “2-blocks with abelian defect groups”, Advances in Mathematics, 254 (2014), 706–735 | DOI | MR | Zbl

[9] Eaton C.W., “Morita equivalence classes of 2-blocks of defect three”, Proc. American Math. Soc., 144 (2016), 1961–1970 | DOI | MR | Zbl

[10] Eaton C.W., Morita equivalence classes of blocks with elementary abelian defect groups of order 16, 2017, 17 pp., arXiv: math.1612.03485

[11] Eaton C.W., Livesey M., “Some examples of Picard groups of blocks”, J. Algebra, 558 (2020), 350–370 | DOI | MR | Zbl

[12] Eaton C.W., Livesey M., “Donovan's conjecture and blocks with abelian defect groups”, Proc. American Math. Soc., 147:3 (2019), 963–970 | DOI | MR | Zbl

[13] Eaton C.W., Eisele F., Livesey M., “Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings”, Math. Z., 295:1 (2020), 249–264 | DOI | MR | Zbl

[14] Eaton C.W. et al., Block library, [e-resource], 2019 URL: https://wiki.manchester.ac.uk/blocks/

[15] Eisele F., On the geometry of lattices and finiteness of Picard groups, 2019, 12 pp., arXiv: math.1908.00129

[16] Erdmann K., “Blocks whose defect groups are Klein four groups: a correction”, J. Algebra, 76:2 (1982), 505–518 | DOI | MR | Zbl

[17] Fong P., Harris M.E., “On perfect isometries and isotypies in finite groups”, Invent Math., 114:1 (1993), 139–191 | DOI | MR | Zbl

[18] Kessar R., Koshitani S., Linckelmann M., “Conjectures of Alperin and Broue for 2-blocks with elementary abelian defect groups of order 8”, Journal fur die reine und angewandte Mathematik, 671 (2012), 85–130 | DOI | MR | Zbl

[19] Kulshammer B., “Crossed products and blocks with normal defect groups”, Communications in Algebra, 13 (1985), 147–168 | DOI | MR | Zbl

[20] Kulshammer B., “Donovan's conjecture, crossed products and algebraic group actions”, Israel J. Math., 92:1 (1995), 295–306 | DOI | MR | Zbl

[21] Kulshammer B., Puig L., “Extensions of nilpotent blocks”, Invent. Math., 102 (1990), 17–71 | DOI | MR | Zbl

[22] Linckelmann M., “The source algebras of blocks with a Klein four defect group”, J. Algebra, 167:3 (1994), 821–854 | DOI | MR | Zbl

[23] Linckelmann M., The block theory of finite group algebras, v. 1 and 2, London Mathematical Society Student Texts, 91-92, Cambridge University Press, Cambridge, 2018, 1114 pp. | MR | Zbl

[24] Linckelmann M., “On automorphisms and focal subgroups of blocks”, Geometric and Topological Aspects of the Representation Theory of Finite Groups, Springer Proceedings in Mathematics Statistics, 242, eds. Carlson J., Iyengar S., Pevtsova J., Springer, Cham, 2018, 235–249 | DOI | MR

[25] Livesey M., On Picard groups of blocks with normal defect groups, 2019, 28 pp., arXiv: math.1907.12167 | MR

[26] McKernon E., “2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle”, J. Algebra, 563 (2020), 30–48 | DOI | MR | Zbl

[27] Murai M., “On blocks of normal subgroups of finite groups”, Osaka J. Math., 50 (2013), 1007–1020 | MR | Zbl

[28] Puig L., “Nilpotent blocks and their source algebras”, Invent. Math., 93:1 (1988), 77–116 | DOI | MR | Zbl

[29] Puig L., Usami Y., “Perfect isometries for blocks with abelian defect groups and Klein four inertial quotients”, J. Algebra, 160:1 (1993), 192–225 | DOI | MR | Zbl

[30] Puig L., “Nilpotent extensions of blocks”, Math. Z., 269:1 (2011), 115–136 | DOI | MR | Zbl

[31] Usami Y., “Perfect isometries for principal Blocks with abelian defect groups and elementary abelian 2-inertial quotients”, J. Algebra, 196:2 (1997), 646–681 | DOI | MR | Zbl

[32] Ward H.N., “On Ree's series of simple groups”, Transactions American Math. Soc., 121:1 (1966), 62–89 | DOI | MR | Zbl

[33] Walter J.H., “The characterization of finite groups with abelian Sylow 2-subgroups”, Annals Math., 89:3 (1969), 405–514 | DOI | MR | Zbl

[34] Zhou Y., “On the $p'$-extensions of inertial blocks”, Proc. American Math. Soc., 144:1 (2016), 41–54 | DOI | MR | Zbl