GenericDecMats

This repository gives access to the generic decomposition matrices of various groups of Lie type.

Currently the following use cases are supported.

• The Gapjm.jl package provides functionality for generic decomposition matrices.

• The generic decomposition matrices can be read into Oscar.jl, its matrices and polynomials are used then.

• There is also a GAP 4 interface, read init.g into a GAP session in order to provide functions for loading, displaying, and testing the matrices.

The interfaces to Gapjm.jl and Oscar.jl can be used in the same Julia session.

There is a preliminary printout version of the data; both the location and the format will be changed.

The data (in the data subdirectory) have been provided by Gunter Malle.

References

[CDR20] Craven, D. A., Dudas, O. and Rouquier, R., Brauer trees of unipotent blocks, J. Eur. Math. Soc. (JEMS), 22 (9) (2020), 2821–2877.

[Dud13] Dudas, O., A note on decomposition numbers for groups of Lie type of small rank, J. Algebra, 388 (2013), 364–373.

[DM15] Dudas, O. and Malle, G., Decomposition matrices for low-rank unitary groups, Proc. Lond. Math. Soc. (3), 110 (6) (2015), 1517–1557.

[DM16] Dudas, O. and Malle, G., Decomposition matrices for exceptional groups at d=4, J. Pure Appl. Algebra, 220 (3) (2016), 1096–1121.

[DM19] Dudas, O. and Malle, G., Bounding Harish-Chandra series, Trans. Amer. Math. Soc., 371 (9) (2019), 6511–6530.

[DM] Dudas, O. and Malle, G., Decomposition matrices for groups of Lie type in non-defining characteristic, arXiv:2001.06395.

[DR14] Dudas, O. and Rouquier, R., Coxeter orbits and Brauer trees III, J. Amer. Math. Soc., 27 (4) (2014), 1117–1145.

[FS84] Fong, P. and Srinivasan, B., Brauer trees in GL(n,q), Math. Z., 187 (1) (1984), 81–88.

[FS90] Fong, P. and Srinivasan, B., Brauer trees in classical groups, J. Algebra, 131 (1) (1990), 179–225.

[Gec91] Geck, M., Generalized Gelʹfand-Graev characters for Steinberg's triality groups and their applications, Comm. Algebra, 19 (12) (1991), 3249–3269.

[GH97] Geck, M. and Hiss, G., Modular representations of finite groups of Lie type in non-defining characteristic, in Finite reductive groups (Luminy, 1994), Birkhäuser Boston, Boston, MA, Progr. Math., 141 (1997), 195–249.

[GHM94] Geck, M., Hiss, G. and Malle, G., Cuspidal unipotent Brauer characters, J. Algebra, 168 (1) (1994), 182–220.

[HN14] Himstedt, F. and Noeske, F., Decomposition numbers of SO_7(q) and Sp_6(q), J. Algebra, 413 (2014), 15–40.

[HL98] Hiss, G. and Lübeck, F., The Brauer trees of the exceptional Chevalley groups of types F_4 and ^2E_6, Arch. Math. (Basel), 70 (1) (1998), 16–21.

[HLM95] Hiss, G., Lübeck, F. and Malle, G., The Brauer trees of the exceptional Chevalley groups of type E_6, Manuscripta Math., 87 (1) (1995), 131–144.

[Jam90] James, G., The decomposition matrices of GL_n(q) for n ≤ 10, Proc. London Math. Soc. (3), 60 (2) (1990), 225–265.

[Mal] Malle, G., Computed directly (in principle known from MR1031453 [Jam90]).

[Miy08] Miyachi, H., Rouquier blocks in Chevalley groups of type E, Adv. Math., 217 (6) (2008), 2841–2871.

[OW98] Okuyama, T. and Waki, K., Decomposition numbers of Sp(4,q), J. Algebra, 199 (2) (1998), 544–555.

[OW02] Okuyama, T. and Waki, K., Decomposition numbers of SU(3,q^2), J. Algebra, 255 (2) (2002), 258–270.

[Sha89] Shamash, J., Brauer trees for blocks of cyclic defect in the groups G_2(q) for primes dividing q^2 ± q + 1, J. Algebra, 123 (2) (1989), 378–396.