Radical subgroups of the finite exceptional groups of Lie typeE6

On the inductive blockwise Alperin weight condition for the unipotent blocks of finite groups of Lie type E_6

“…according to the proof of [3, Theorem 3.6] and [3,Lemma 3.5]. Note that the restriction of the diagonal automorphism, the field automorphism and the graph automorphism to N G (R 5 ) is exactly the diagonal automorphism, the field automorphism and the graph automorphism of N G (R 5 ) respectively.…”

“…In this article, we consider finite exceptional types of Lie type E 6 and 2 E 6 based on the classification of their radical subgroups in [3]. We write E +1 6 (q) = E 6 (q) and E −1 6 (q) = 2 E 6 (q) for the universal versions.…”

“…Following the notation in [3], for finite groups A, B, we write A.B for an extension of A by B. If n, m are positive integers, we write n m for the direct product of m copies of cyclic groups of order n. For η ∈ {±1}, we define…”

“…Proof. By [15,Table 2], T 1 = (q+ε) 2 ×(q 2 −1) 2 , T 2 = (q 2 +εq+1) 3 , T 3 = ((q 2 +1)(q−ε)) 2 and T 4 = (q 2 − εq + 1) × (q 4 + q 2 + 1) are maximal tori of G. Moreover, N G (T 1 )/T 1 2 1+4…”

“…in[3, Table 3]). Since there is only one defect zero character of SL 2(5) by [13, p.2], we can see that there are 8 unipotent weights afforded by R 19 and 2 unipotent weights afforded by R 21 and others afford no unipotent weights.…”