| Frank Lübeck |
Tables of Green Functions for Exceptional Groups
On this page we provide data files containing the generalized Green functions for untwisted finite groups of Lie type coming from adjoint simple algebraic groups of exceptional type.
The files below can be read into both GAP3 and
GAP4.
The data are given in terms of a parameter q which describes
the order of the finite field over which a group is defined.
If in your GAP session the variable q is bound before reading
one of these data files, then q will be specialized to this
value when reading the data. If q is unbound it will be
assigned an indeterminate over the Cyclotomics with name
"q".
The data are stored in GAP records and can be accessed as follows:
.table[i][j]- values of Green functions as polynomials in q. The indices i correspond to the Green functions and j to the unipotent classes.
.charInfo[i]- labeling of Green function number i. This is a pair describing the type of a Levi subgroup ("A_0" corresponds to the ordinary Green functions) and a label for the conjugacy class in the relative Weyl group.
.classInfo[j]- description of unipotent class number j. Given as a pair, a label for the geometric class and a counter for the rational classes.
.classLengths[j]- size of unipotent class number j. Given as polynomial in
q. .GFOrder- order of the group. Also as polynomial in
q.
The Data
Click on the filenames to view or download the data. The right colums
gives the names of the GAP records containing the data. In type E8 there
are two such records whose data are relevant if q is congruent
to 1 or 2 modulo 3, respectively.
| Filename | record name(s) | |
|---|---|---|
| GreenG2good.g | greenG2good | |
| GreenG2char2.g | greenG2char2 | |
| GreenG2char3.g | greenG2char3 | |
| GreenF4good.g | greenF4good | |
| GreenF4char2.g | greenF4char2 | |
| GreenF4char3.g | greenF4char3 | |
| GreenE6good.g | greenE6good | |
| GreenE6char2.g | greenE6char2 | |
| GreenE6char3.g | greenE6char3 | |
| GreenE7good.g | greenE7good | |
| GreenE7char2.g | greenE7char2 | |
| GreenE7char3.g | greenE7char3 | |
| GreenE8good.g | greenE8good1mod3, greenE8good2mod3 | |
| GreenE8char2_1mod3.g | greenE8char2_1mod3 | |
| GreenE8char2_2mod3.g | greenE8char2_2mod3 | |
| GreenE8char3.g | greenE8char3 | |
| GreenE8char5_1mod3.g | greenE8char5_1mod3 | |
| GreenE8char5_2mod3.g | greenE8char5_2mod3 |
References
These data were (re-)computed using the algorithm given in [5, Section 24]. The input is the generalized Springer correspondence given in [9] and character tables of Weyl groups from [1]. In fact, the mentioned algorithm from [5] does not yield the (generalized) Green functions as class functions but as linear combinations of certain functions which have support on the rational classes corresponding to one geometric unipotent class. For the values of these functions in good characteristic we are implicitly using the original references where all these tables were determined for the first time. For more details see the survey [8] and the references given there.
The bad characteristic cases were done later by various authors and we add the references here.
[1] Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G.,
CHEVIE---a system for computing and processing generic
character tables,
Appl. Algebra Engrg. Comm. Comput.,
7 (3)
(1996),
175--210
(Computational methods in Lie theory (Essen, 1994)).
[2] Geck, M., Computing Green functions in small characteristic, J. Algebra, 561, (2020), 163--199.
[3] Lübeck, F., Green functions in small characteristic, arXiv preprint, 2403.18190, (2024).
[4] Lübeck, F., Shoji, T., Generalized Green functions and unipotent classes for finite reductive groups, IV, arXiv preprint, 2408.16960, (2024).
[5] Lusztig, G., Character sheaves. V, Adv. in Math., 61 (2) (1986), 103--155.
[6] Malle, G., Green functions for groups of types E6 and F4 in characteristic 2, Comm. Algebra, 21, (1993), 747--798.
[7] Porsch, U., Die Greenfunktionen der endlichen Gruppen E6(q), q=3n, Universität Heidelberg, Diploma thesis, (1994).
[8] Shoji, T., Green functions of reductive groups over a finite field, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Amer. Math. Soc., Proc. Sympos. Pure Math., 47, Providence, RI (1987), 289--301.
[9] Spaltenstein, N., On the generalized Springer correspondence for exceptional groups, in Algebraic groups and related topics (Kyoto/Nagoya, 1983), North-Holland, Adv. Stud. Pure Math., 6, Amsterdam (1985), 317--338.
Zuletzt geändert: Do 12.12.2024, 23:00:03 (UTC)