Page 610 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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ALGEBRA
Domination of blocks, fusion systems and hyperfocal subgroups
Tiberiu Coconet, tiberiu.coconet@econ.ubbcluj.ro
Babes-Bolyai University Cluj-Napoca, Romania
Coauthor: Constantin-Cosmin Todea
In the context of modular representation theory of finite groups, considering a finite group
G, an algebraically closed field k of characteristic p, a block b of kG and a maximal Brauer
(D, e), the block b is inertial if b and e lie in a special type of Morita equivalence. A particular
situation of this equivalence makes b into a nilpotent block. For a normal p-subgroup P of
G, setting G¯ := G/P, the G-acted epimorphism of group algebras π : kG → kG¯ determines
the connection between b and its dominating blocks. We investigate the connections between
some properties of blocks and of their dominating blocks. We find conditions to verify that a
block is inertial if and only if its dominating block is inertial. In some situations the equality of
the factor fusion systems associated with a block and with its Brauer correspondent block give
information about the hyperfocal subgroups.
References
[1] CRAVEN, D., The theory of fusion systems, vol.1, Cambridge University Press, 2011.
[2] CRAVEN, D., Control of fusion and solubility in fusion system, J. Algebra, 323 (2010),
2429–2448.
[3] EATON, C. W. and LIVESEY, M., Donovan’s conjecture and blocks with abelian defect
groups, Proc. Amer. Math. Soc., 147 (2019), 963–970.
[4] FEIT, W., The representation theory of finite groups, North-Holland Publishing Company,
Cambridge, Amsterdam New York Oxford, 1982.
[5] LINCKELMANN, M., The block theory of finite group algebras, Volume1, London Math-
ematical Society Student Texts, Cambridge, 2018.
[6] NAGAO, H. and TSUSHIMA, Y., Representations of Finite Groups, Academic Press, Lon-
don, 1989.
[7] THÉVENAZ, J., G-Algebras and Modular Representation Theory, Clarendon Press, Ox-
ford 1995.
[8] WATANABE, A., On nilpotent blocks of finite groups, J. Algebra, 163 (1990), 128–134.
[9] WATANABE, A., On blocks of finite groups with central hyperfocal subgroups, J. Algebra
368 (2012) 358—375.
[10] YUN, F., Hyperfocal subalgebras of blocks and computation of characters, J. Algebra 322
(2009) 3681–3692.
608
Domination of blocks, fusion systems and hyperfocal subgroups
Tiberiu Coconet, tiberiu.coconet@econ.ubbcluj.ro
Babes-Bolyai University Cluj-Napoca, Romania
Coauthor: Constantin-Cosmin Todea
In the context of modular representation theory of finite groups, considering a finite group
G, an algebraically closed field k of characteristic p, a block b of kG and a maximal Brauer
(D, e), the block b is inertial if b and e lie in a special type of Morita equivalence. A particular
situation of this equivalence makes b into a nilpotent block. For a normal p-subgroup P of
G, setting G¯ := G/P, the G-acted epimorphism of group algebras π : kG → kG¯ determines
the connection between b and its dominating blocks. We investigate the connections between
some properties of blocks and of their dominating blocks. We find conditions to verify that a
block is inertial if and only if its dominating block is inertial. In some situations the equality of
the factor fusion systems associated with a block and with its Brauer correspondent block give
information about the hyperfocal subgroups.
References
[1] CRAVEN, D., The theory of fusion systems, vol.1, Cambridge University Press, 2011.
[2] CRAVEN, D., Control of fusion and solubility in fusion system, J. Algebra, 323 (2010),
2429–2448.
[3] EATON, C. W. and LIVESEY, M., Donovan’s conjecture and blocks with abelian defect
groups, Proc. Amer. Math. Soc., 147 (2019), 963–970.
[4] FEIT, W., The representation theory of finite groups, North-Holland Publishing Company,
Cambridge, Amsterdam New York Oxford, 1982.
[5] LINCKELMANN, M., The block theory of finite group algebras, Volume1, London Math-
ematical Society Student Texts, Cambridge, 2018.
[6] NAGAO, H. and TSUSHIMA, Y., Representations of Finite Groups, Academic Press, Lon-
don, 1989.
[7] THÉVENAZ, J., G-Algebras and Modular Representation Theory, Clarendon Press, Ox-
ford 1995.
[8] WATANABE, A., On nilpotent blocks of finite groups, J. Algebra, 163 (1990), 128–134.
[9] WATANABE, A., On blocks of finite groups with central hyperfocal subgroups, J. Algebra
368 (2012) 358—375.
[10] YUN, F., Hyperfocal subalgebras of blocks and computation of characters, J. Algebra 322
(2009) 3681–3692.
608
