Cross-Characteristic Character and Fixed Point Space Ratios for Groups of Lie Type

✍ Scribed by David Gluck; Kay Magaard

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 208 KB
• Volume: 204
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7277

✦ Synopsis

Let H be a finite group such that F* HrZ H is a simple group of Lie type of rank l defined in characteristic p. Let K be a field and let V w x be a faithful irreducible K H -module. A recent result of Hall, Liebeck, w x ࠻ Ž . Ž and Seitz 8, Theorem 4a shows that, for all

. y 1 . 16 l q 8 dim V. Ž . If char K s p, then this bound has the correct order of magnitude. If Ž . Ž . char K / p, however, one might expect that dim C x rdim V is bounded V away from 1, independently of l. We establish such a uniform bound Ž . in Corollary 3.6 below; we show that dim C x rdim V F 71r72 for all V x g H ࠻ . Recent work on estimating the minimal base size of a primitive affine permutation group provides one motivation for proving this uniform bound, but other applications are expected.

Our result depends on the bounds we establish in Theorem 3.1 for Brauer character ratios of groups of Lie type in non-defining characteristic. If G is a group of Lie type defined in characteristic p, and is an irreducible r-Brauer character of G for a prime r / p, we obtain weak < Ž . Ž .< upper bounds for x r 1 when x g G is a nonidentity unipotent element or when x is a noncentral r Ј-element which lies in some proper Ž .

' parabolic of G. For unipotent elements we obtain 1r2 q O 1r q bounds < Ž . Ž .< for x r 1 , while for semisimple elements in proper parabolics we < Ž . Ž .< Ž . show that x r 1 F 1r2 q O 1rq . For both types of elements, < Ž . Ž .<

x r 1 F 6r7 for all q. < Ž . Ž .< These bounds for x r 1 are weaker than those obtained for w x ordinary characters in 4 . There are several reasons for this. First of *Research partially supported by an NSA grant. 188