The invariant rings of the Sylow groups of GU(3,q2), GU(4,q2), Sp(4,q) and O+(4,q) in the natural characteristic

Abstract

Let G be a Sylow p-subgroup of the unitary groups GU (3, q 2 ), GU (4, q 2 ), the sympletic group Sp(4, q) and, for q odd, the orthogonal group O + (4, q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.

Modular invariants of finite gluing groups

“…The above is consistent with the calculation of F q [V ] USp 4 in [25]. Since USp 2m is a Sylow psubgroup for all of the standard parabolic subgroups of Sp 2m (F q ), the following is a consequence of [10].…”

Modular Invariants of Finite Affine Linear Groups

“…The above is consistent with the calculation of F q [V ] USp 4 in [25]. Since U Sp 2m is a Sylow psubgroup for all of the standard parabolic subgroups of Sp 2m (F q ), the following is a consequence of [10].…”

“…Then E is a Sylow p-subgroup for O + 4 (F q ). The ring of invariants F q [W 2 ] E is a complete intersection generated by u, an element in degree q + 1 constructed by applying a Steenrod operation to u, and the orbit products of the variables: N E (x 4 ) of degree q 3 , N E (x 2 ) and N E (x 2 ) of degree q, and x 1 (see [25]). Let E 1 denote the subgroup of E of order q corresponding to taking c 2 = 0.…”