Projective representations for some exceptional finite groups of Lie type. (English) Zbl 1004.20004
Collins, Michael J. (ed.) et al., Modular representation theory of finite groups. Proceedings of a symposium, University of Virginia, Charlottesville, VA, USA, May 8-15, 1998. Berlin: de Gruyter. 223-230 (2001).
Let \(G\) be a finite simple group and let \(d_l(G)\) be the smallest degree of nontrivial projective representations of \(G\) over a field of characteristic \(l\). Lower bounds for \(d_l(G)\) were found by V. Landazuri and G. M. Seitz [J. Algebra 32, 418-443 (1974; Zbl 0325.20008)] and refined by G. M. Seitz and A. E. Zalesskij [J. Algebra 158, No. 1, 233-243 (1993; Zbl 0789.20014)]. These bounds are actually the best possible ones in many cases and they have been used in numerous applications.
In some cases they can still be improved. If \(G\) is a finite classical group, then these are the cases of \(\text{PSL}_n(q)\) and orthogonal groups, and the bounds have been improved by R. Guralnick, T. Penttila, C. E. Praeger, and J. Saxl [Proc. Lond. Math. Soc., III. Ser. 78, No. 1, 167-214 (1999; Zbl 1041.20035)] and by the author [in J. Algebra 229, No. 2, 666-677 (2000; Zbl 0962.20034)], respectively.
This paper refines the bounds for exceptional groups of types \(E_6\), \(E_7\), and \(E_8\). Notice that the smallest complex degree \(d_0(G)\) for these groups has been computed by F. Lübeck [in Commun. Algebra 29, No. 5, 2147-2169 (2001; see the preceding review Zbl 1004.20003)]. The author proves that \(d_l(G)\geq d_0(G)-3\) for these groups, and this new bound is very close to actually give the precise value of \(d_l(G)\).
The Landazuri-Seitz-Zalesskij bounds have now been improved also for the groups of types \(^3D_4\) and \(^2E_6\) by K. Magaard, G. Malle, and the reviewer [in Pac. J. Math. 202, No. 2, 379-427 (2002; Zbl 1072.20013)].
For the entire collection see [Zbl 0977.00027].
In some cases they can still be improved. If \(G\) is a finite classical group, then these are the cases of \(\text{PSL}_n(q)\) and orthogonal groups, and the bounds have been improved by R. Guralnick, T. Penttila, C. E. Praeger, and J. Saxl [Proc. Lond. Math. Soc., III. Ser. 78, No. 1, 167-214 (1999; Zbl 1041.20035)] and by the author [in J. Algebra 229, No. 2, 666-677 (2000; Zbl 0962.20034)], respectively.
This paper refines the bounds for exceptional groups of types \(E_6\), \(E_7\), and \(E_8\). Notice that the smallest complex degree \(d_0(G)\) for these groups has been computed by F. Lübeck [in Commun. Algebra 29, No. 5, 2147-2169 (2001; see the preceding review Zbl 1004.20003)]. The author proves that \(d_l(G)\geq d_0(G)-3\) for these groups, and this new bound is very close to actually give the precise value of \(d_l(G)\).
The Landazuri-Seitz-Zalesskij bounds have now been improved also for the groups of types \(^3D_4\) and \(^2E_6\) by K. Magaard, G. Malle, and the reviewer [in Pac. J. Math. 202, No. 2, 379-427 (2002; Zbl 1072.20013)].
For the entire collection see [Zbl 0977.00027].
Reviewer: Pham Huu Tiep (Gainesville)
MSC:
| 20C33 | Representations of finite groups of Lie type |
| 20C25 | Projective representations and multipliers |
