Group actions of some subgroups of parabolic subgroups
Journal
Journal of Algebra
Journal Volume
185
Journal Issue
1
Pages
175-183
Date Issued
1996
Author(s)
Abstract
Let K be the finite field with q elements, V an n-dimensional vector space over K, P the parabolic subgroup of GL(V) associated to the flag V := V0 ⊃ V1 ⊃ ⋯ ⊃ Vr = {0}, U the unipotent radical of P. Suppose that G is a subgroup of GL(V) such that (i) U ⊂ G ⊂ P, and (ii) Image {G → GL(V0/V1) × GL(V1/V2) × ⋯ × GL(Vr-1/Vr)} = G1 × G2 × ⋯ × Gr, where Gi is a subgroup of GL(Vi-1/Vi) for 1 ≤ i ≤ r. THEOREM, (i) If K[Vi-1/Vi]Gi is a polynomial ring (resp. a complete intersection, a Gorenstein ring, a Cohen-Macaulay ring) for 1 ≤ i ≤ r, so is K[V]G. (ii) If K[Vi- /Vi] is a free module over K[Vi-1/Vi]Gi for 1 ≤ i ≤ r, so is K[V] over K[V]G. (iii) If K(Vi-1/Vi)Gi is rational over K for 1 ≤ i ≤ r, so is K(V)G over K. © 1996 Academic Press, Inc.
Type
journal article