scholarly journals CHARACTER LEVELS AND CHARACTER BOUNDS

2020 ◽  
Vol 8 ◽  
Author(s):  
ROBERT M. GURALNICK ◽  
MICHAEL LARSEN ◽  
PHAM HUU TIEP
Keyword(s):  
Random Walks ◽  
Mixing Time ◽  
Upper Bounds ◽  
Unitary Groups ◽  
Covering Number ◽  
Classical Groups ◽  
Dual Pairs ◽  
Special Linear ◽  
Exponential Character ◽  
Finite Classical Groups

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.

Conjugacy classes of small sizes in the linear and unitary groups

2013 ◽  
Vol 16 (6) ◽  
Author(s):  
Shawn T. Burkett ◽  
Hung Ngoc Nguyen
Keyword(s):  
Conjugacy Classes ◽  
Unitary Groups ◽  
General Linear ◽  
Linear Groups ◽  
Classical Groups ◽  
General Linear Groups ◽  
Finite Classical Groups

Abstract.Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in

scholarly journals Non-Hurwitz Classical Groups

2007 ◽  
Vol 10 ◽  
pp. 21-82 ◽  
Author(s):  
R. Vincent ◽  
A.E. Zalesski
Keyword(s):  
Finite Fields ◽  
Unitary Groups ◽  
Classical Groups ◽  
Special Linear ◽  
Low Dimensional

AbstractIn previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra28 (2000) 5383–5404] it is shown that certain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the special linear and special unitary groups in dimensions 8.9,11.13. We also show that all groups Sp(n, q) are not Hurwitz forqeven andn= 6,8,12,16. In the range 11 <n< 32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3),PΩ±(8, 3k),PΩ±10k), Ω(11,3k), Ω±(14,3k), Ω±(16,7k), Ω(n, 7k) forn= 9,11,13, PSp(8, 7k) are not Hurwitz.

scholarly journals Power series coefficients for probabilities in finite classical groups

2006 ◽  
Vol 305 (2) ◽  
pp. 1212-1237
Author(s):  
John R. Britnell ◽  
Jason Fulman
Keyword(s):  
Power Series ◽  
Classical Groups ◽  
Finite Classical Groups

scholarly journals The sylow 2-subgroups of the finite classical groups

1964 ◽  
Vol 1 (2) ◽  
pp. 139-151 ◽  
Author(s):  
Roger Carter ◽  
Paul Fong
Keyword(s):  
Classical Groups ◽  
Finite Classical Groups

RANDOM (r, s)-GENERATION OF FINITE CLASSICAL GROUPS

2002 ◽  
Vol 34 (2) ◽  
pp. 185-188 ◽  
Author(s):  
MARTIN W. LIEBECK ◽  
ANER SHALEV
Keyword(s):  
Classical Groups ◽  
Large Rank ◽  
Finite Classical Groups

A proof is given that for primes r, s, not both 2, and for finite simple classical groups G of sufficiently large rank, the probability that two randomly chosen elements in G of orders r and s generate G tends to 1 as |G| → ∞.

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