On the character degrees of Sylow p-subgroups of Chevalley groups G(p(f)) of type E

摘要

Let F-q be a field of characteristic p with q elements. It is known that the degrees of the irreducible characters of the Sylow p-subgroup of GL(F-q)are powers of q. On the other hand Sangroniz (2003) showed that this is true for a Sylow p-subgroup of a classical group defined over F-q if and only if p is odd. For the classical groups of Lie type B, C and D the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow p-subgroups of the Chevalley groups D-4(q)with q = 2(f) of degree q(3)/2. Then we use an analogous construction for E-6(q) with q = 3(f) to obtain characters of degree q(7)/3, and for E-8(q) with q = 5(f) to obtain characters of degree q(16)/5. This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type E in terms of the representation theory of the Sylow p-subgroup.