Pham Huu Tiep

Real conjugacy classes in algebraic groups and finite groups of Lie type

According to the Berman–Witt theorem, the number of real classes of G is equal to the number of complex irreducible characters whose values are real (such characters are called real ). Each real irreducible character is the character of a real or quaternion representation of G. For this reason Problem 1.1 has attracted considerable attention for various classes of groups; see [9], [12], [13]. In particular, Feit and Zuckerman [9] studied this problem for classical groups extended by a graph automorphism. They also showed that all elements are real in the groups Sp2nðqÞ with q1 1 ðmod 4Þ, and Gow [12] proved this for q even. There are some other results in the literature which are not concerned with quasi-simple finite groups. In particular, Gow [13] proved that all elements in SOnðqÞ are real for n1 0 ðmod 4Þ and for n odd. This is also true for GOnðqÞ with n arbitrary; see [8], [13], [20]. The main result of the paper is the following theorem which completely solves Problem 1.1 for finite quasi-simple groups:

Low-Dimensional Representations of Special Linear Groups in Cross Characteristics

The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group

Rational irreducible characters and rational conjugacy classes in finite groups

\mbox{PSL}_{n}(q)

Cross characteristic representations of even characteristic symplectic groups

are investigated. If

Prime divisors of character degrees

n \geq 3

On restrictions of modular spin representations of symmetric and alternating groups

and

Decompositions of small tensor powers and Larsen’s conjecture

(n,q) \neq (3,2)

Globally Irreducible Representations of Finite Groups and Integral Lattices

,

Brauer's height zero conjecture for the 2-blocks of maximal defect

(3,4)

Representations of the general linear groups which are irreducible over subgroups

,