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

Abstract

Let 𝔽

On the characters of Sylow $p$-subgroups of finite Chevalley groups $G(p^f)$ for arbitrary primes

“…Hence we have X = {x 2 (t 2 )x 5 (t 5 ) | t 2 , t 5 ∈ F q , a 2 8 t 2 5 = a 9 t 2 and a 2 10 t 2 2 = a 18 t 5 } = {x 2 (t 2 )x 5 (t 5 ) | t 2 , t 5 ∈ F q , t 2 = a 2 8 a −1 9 t 2 5 and t 4 5 = a −4 8 a 2 9 a −2 10 a 18 t 5 } and Y = {x 3 (s 3 )x 7 (s 7 ) | s 3 , s 7 ∈ F q , a 9 s 2 3 = a 10 s 7 and a 18 s 2 7 = a 8 s 3 } = {x 3 (s 3 )x 7 (s 7 ) | s 3 , s 7 ∈ F q , s 7 = a 9 a −1 10 s 2 3 and s 4 3 = a 8 a −2 9 a 2 10 a −1 18 s 3 }. 15 Let us assume that f = 2k. If a 18 / ∈ a 8 a −2 9 a 2 10 F × q,3 , that is, for 2(q − 1)/3 choices of a 18 in F × q , then the quartic equations involved in the definitions of X and Y just have a trivial solution.…”

The generic character table of a Sylow p-subgroup of a finite Chevalley group of type D4

“…Then we define χ a 8 ,a 9 ,a 10 ,a 5,6,7 ,d 1,2,4 ,d 3 8,9,10,q 3 /2 by tensoring ψ d 1,2,4 with χ 0,0,d 3 ,0 lin . The proposition below states the values of these characters, these can be deduced from [30,Thm. 2.3].…”

“…2.3]. Note that with our specific choice of φ, we have ker [30,Definition 1.2]. In turn this implies that a φ as defined in [30,Definition 1.4] is equal to a −p .…”

“…Note that with our specific choice of φ, we have ker [30,Definition 1.2]. In turn this implies that a φ as defined in [30,Definition 1.4] is equal to a −p . We note that the values for χ a 8 ,a 9 ,a 10 8,9,10,q 3 look different to those for p > 2, which is explained by the fact that the quadratic Gauss sums for p = 2 are clearly zero.…”

“…to be the character obtained by inducing λ a 11 Using [30,Lemma 1.5] with Z = X 11 , Y = X 8 X 9 X 10 , X = X 1 X 2 X 4 and M = X 3 X 5 X 6 X 7 , we obtain that each χ a 11 ,b 5 ,b 6 ,b 7 ,b 3…”