The ranks of primitive parabolic permutation representations of the simple groups B l (q), C l (q), and D l (q)

Abstract

We determine the ranks of the permutation representations of the simple groups B l (q), C l (q), and D l (q) on the cosets of the parabolic maximal subgroups.

Primitive parabolic permutation representations for finite symplectic groups

“…In compliance with this notation, {G W } is a trivial suborbit, whose subdegree is equal to 1. If dim W = 1 then a permutation representation has minimal degree (see [1] for details). Denote by X : Y the split extension of a group X by a group Y , and by [b] an arbitrary soluble group of order b, b ∈ N. We write E j for an identity matrix of order j, and A t for the transpose of a matrix A.…”

“…An important class of permutation representations for these groups is formed by parabolic representations, i.e., representations on cosets with respect to parabolic subgroups. A brief review of results on primitive parabolic permutation representations for finite groups of Lie type is contained in [1].…”

“…In [2][3][4][5][6], these parameters were determined for primitive parabolic permutation representations of all finite simple exceptional groups of Lie type. Ranks of permutation representations for classical groups A l (q), B l (q), C l (q), and D l (q) on cosets with respect to maximal parabolic subgroups were computed in [1,7]. In [8], subdegrees of groups A l (q) were determined, and it was shown how the point stabilizer and double stabilizers for such groups are structured.…”

“…Ranks of permutation representations for classical groups A l (q), B l (q), C l (q), and D l (q) on cosets with respect to maximal parabolic subgroups were computed in [1,7]. In [8], subdegrees of groups A l (q) were determined, and it was shown how the point stabilizer and double stabilizers for such groups are structured. In [1,7,8], proofs for the classical groups in question were effected in terms of root systems.…”

“…In [8], subdegrees of groups A l (q) were determined, and it was shown how the point stabilizer and double stabilizers for such groups are structured. In [1,7,8], proofs for the classical groups in question were effected in terms of root systems. Parameters of primitive parabolic permutation representations of finite special linear and unitary groups were determined in [9], in which proofs were based on using linear transformations and bilinear forms.…”