Weights for Classical Groups

Abstract

Abstract.This paper proves the Alperin's weight conjecture for the finite unitary groups when the characteristic r of modular representation is odd. Moreover, this paper proves the conjecture for finite odd dimensional special orthogonal groups and gives a combinatorial way to count the number of weights, block by block, for finite symplectic and even dimensional special orthogonal groups when r and the defining characteristic of the groups are odd.

Nilpotent blocks of quasisimple groups for odd primes

“…There are many examples of non-nilpotent blocks covering nilpotent blocks, but there are also examples of nilpotent blocks covering non-nilpotent blocks, such as the following (which came to light during a conversation with Radha Kessar): Example 2.3 Let G = PGL (3,7), N = P SL(3, 7) and p = 2, so that [G:N ] = 3. Then N has a unique block b with defect group D ∼ = Z 2 × Z 2 and b is not nilpotent.…”

“…Let V be a linear, unitary, non-degenerate orthogonal or symplectic space over the field F q , where q = r a for some prime r = p. We will follow the notation of [3], [11], [16] and [17].…”

“…[3, p.6]). For Γ ∈ F q (G), let d Γ be the degree of Γ, and δ Γ be the reduced degree defined as in [3], [16] and [17].…”

Nilpotent Blocks of Quasisimple Groups for the Prime Two

“…The next lemma, proved in [6], says that (Q, b Q ) must be a B-subgroup, and that dominance of subpairs respects the usual partial order on B-subgroups: 3 The symmetric and alternating groups WriteŜ n for the double cover of the symmetric group S n . Then the 2-blocks ofŜ n and of S n are in one-to-one correspondence under the natural epimorphism, and the block corresponding to a nilpotent block is nilpotent (by [29,Lemma 2]).…”

Dade’s invariant conjecture for general linear and unitary groups in non-defining characteristics

“…In addition, let GL (n, q) = GL(n, q) or U(n, q) according to whether = + or −. The radical subgroups of G are classified by [2] and [3]. We shall follow the notation of [2].…”