Automorphism groups of invariant lattices in the Steinberg module of groups of Lie type of odd characteristic
Sbornik. Mathematics, Tome 189 (1998) no. 9, pp. 1273-1294
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The purpose of this article is to compute the automorphism groups of $G$-invariant lattices in the Steinberg module $V$, where $G$ is a finite simple group of Lie type over a field of characteristic $p>2$.
@article{SM_1998_189_9_a0,
     author = {K. S. Abdukhalikov},
     title = {Automorphism groups of invariant lattices in {the~Steinberg} module of groups of {Lie} type of odd characteristic},
     journal = {Sbornik. Mathematics},
     pages = {1273--1294},
     year = {1998},
     volume = {189},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_9_a0/}
}
TY  - JOUR
AU  - K. S. Abdukhalikov
TI  - Automorphism groups of invariant lattices in the Steinberg module of groups of Lie type of odd characteristic
JO  - Sbornik. Mathematics
PY  - 1998
SP  - 1273
EP  - 1294
VL  - 189
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_1998_189_9_a0/
LA  - en
ID  - SM_1998_189_9_a0
ER  - 
%0 Journal Article
%A K. S. Abdukhalikov
%T Automorphism groups of invariant lattices in the Steinberg module of groups of Lie type of odd characteristic
%J Sbornik. Mathematics
%D 1998
%P 1273-1294
%V 189
%N 9
%U http://geodesic.mathdoc.fr/item/SM_1998_189_9_a0/
%G en
%F SM_1998_189_9_a0
K. S. Abdukhalikov. Automorphism groups of invariant lattices in the Steinberg module of groups of Lie type of odd characteristic. Sbornik. Mathematics, Tome 189 (1998) no. 9, pp. 1273-1294. http://geodesic.mathdoc.fr/item/SM_1998_189_9_a0/

[1] Ivanov D. N., “Ortogonalnye razlozheniya algebr Li tipa $A_{p^n-1}$ i $D_n$ s konechnym chislom klassov podobnykh invariantnykh podreshetok”, Vestn. MGU. Ser. 1. Matem., mekh., 1989, no. 2, 40–43 | MR | Zbl

[2] Abdukhalikov K. S., “Tselochislennye invariantnye reshetki v algebrakh Li tipa $A_{p^m-1}$”, Matem. sb., 184:4 (1993), 61–104 | MR | Zbl

[3] Abdukhalikov K. S., “Tselochislennye reshetki, assotsiirovannye s konechnoi affinnoi gruppoi”, Matem. sb., 185:12 (1994), 3–18 | Zbl

[4] Bondal A. I., Kostrikin A. I., Fam Khyu Tep, “Invariantnye reshetki, reshetka Licha i ee chetnye unimodulyarnye analogi v algebrakh $A_{p-1}$”, Matem. sb., 130 (172):4 (8) (1986), 435–464 | MR | Zbl

[5] Kostrikin A. I., Pham Huu Tiep, Orthogonal decompositions and integral lattices, Walter de Gruyter, Berlin, 1994 | MR

[6] Burichenko V. P., “Invariantnye reshetki v module Steinberga i ikh gruppy izometrii”, Matem. sb., 184:12 (1993), 145–156 | MR | Zbl

[7] Carter R. W., Finite groups of Lie type. Conjugacy classes and complex characters, Wiley and Sons, New York, 1985 | MR | Zbl

[8] Zsigmondy K., “Zur Theorie der Potenzreste”, Monatsh. Math. Phys., 3 (1892), 265–284 | DOI | MR

[9] Kleidman P. B., Liebeck M. W., The subgroup structure of the finite classical groups, Cambridge Univ. Press, Cambridge, 1990 | MR

[10] Lusztig G., Characters of reductive groups over a finite field, Princeton Univ. Press, Princeton, 1984 | MR | Zbl

[11] Lusztig G., “On the representations of reductive groups with disconnected center”, Astérisque, 168 (1988), 157–166 | MR | Zbl

[12] Digne F., Michel J., Representations of finite groups of Lie type, Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[13] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A., Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[14] Guralnick R. M., “Subgroups of prime power index in a simple group”, J. Algebra, 81 (1983), 304–311 | DOI | MR | Zbl

[15] Kertis Ch., Rainer I., Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Nauka, M., 1969 | MR

[16] Dixon J., Mortimer B., “The primitive permutation groups of degree less than 1000”, Math. Proc. Cambridge Philos. Soc., 103 (1988), 213–238 | DOI | MR | Zbl

[17] Wagner A., “An observation on the degrees of projective representations of the symmetric and alternating groups over an arbitrary field”, Arch. Math., 29 (1977), 583–589 | DOI | MR | Zbl

[18] Rasala R., “On the minimal degrees of characteres of $S_n$”, J. Algebra, 45 (1977), 132–181 | DOI | MR | Zbl

[19] Cooperstein B. N., “Minimal degree for a permutation representation of a classical group”, Israel J. Math., 30 (1978), 213–235 | DOI | MR | Zbl

[20] Liebeck M. W., Saxl J., “On the orders of maximal subgroups of the finite exceptional groups of Lie type”, Proc. London Math. Soc. (3), 55 (1987), 299–330 | DOI | MR | Zbl

[21] Liebeck M. W., Saxl J., “Maximal subgroups of finite simple groups and their automorphism groups”, Contemp. Math., 131 (1992), 243–259 | MR | Zbl

[22] Saxl J., “The complex caharacters of the symmetric groups that remains irreducible in subgroups”, J. Algebra, 111 (1987), 210–219 | DOI | MR | Zbl

[23] James G. D., The representation theory of the symmetric groups, Springer-Verlag, Berlin, 1978 | MR | Zbl

[24] Cameron P. J., “Finite permutation groups and finite simple groups”, Bull. London Math. Soc., 13 (1981), 1–22 | DOI | MR | Zbl

[25] Plesken W., “On absolutely irreducible representations of orders”, Number Theory and Algebra, Collect. Rap. dedic. H. B. Mann. A. E. Ross, O. Taussky-Todd, 1977, 241–262 | MR | Zbl

[26] Seitz G. M., “Flag-transitive subgroups of Chevalley groups”, Ann. of Math., 97 (1973), 27–56 | DOI | MR | Zbl

[27] Landazuri V., Seitz G. M., “On the minimal degrees of projective representations of the finite Chevalley groups”, J. Algebra, 32 (1974), 418–443 | DOI | MR | Zbl

[28] Seitz G. M., Zalesskii A. E., “On the minimal degrees of projective representations of the finite Chevalley groups, II”, J. Algebra, 158 (1993), 233–243 | DOI | MR | Zbl

[29] Tiep P. H., Zalesskii A. E., “Minimal characters of the finite classical groups”, Comm. Algebra, 24 (1996), 2093–2167 | DOI | MR | Zbl