The ring of invariants of the orthogonal group over finite fields in odd characteristic

Let $V$ be a non-zero finite dimensional vector space over a finite field \mathbb{F}_q of odd characteristic. Fixing a non-singular quadratic form \xi₀ in S²(V^*), the symmetric square of the dual of V we are concerned with the Orthogonal group O(\xi₀), the subgroup of the General Linear Group $GL(V)$ that fixes \xi₀ and with invariants of this group. We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of O(\xi₀). Considering the O(\xi₀) orbits of the dual vector space $\vs$ we generate the Chern Orbit polynomials, the coefficients of which, the Chern Orbit Classes, are also invariants of the Orthogonal group. The invariants \xi₁, \xi₂, \dots are be generated from \xi₀ by applying the action of the Steenrod Algebra to S²(V^*) which being natural takes invariants to invariants. Our aim is to discover further invariants from these known invariants with the intention of establishing a set of generators for the the Ring of invariants of the Orthogonal Group. In particular we calculate invariants of O(\xi₀) when the dimension of the vector space is $4$ the finite field is \mathbb{F}₃ and the quadratic form is \xi₀=x₁²+x₂²+x₃²+x₄² and we are able to establish an explicit presentation of O(\xi₀) in this case.

Level dc:type.qualificationlevel

PhD

Grantor dc:publisher.institution

University of Glasgow

Year dc:date.issued

2008

Author dc:creator

• Barnes, Sue

• QA Mathematics

Language dc:language

en