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

Barnes, Sue (2008) The ring of invariants of the orthogonal group over finite fields in odd characteristic. PhD thesis, University of Glasgow.

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Printed Thesis Information: https://eleanor.lib.gla.ac.uk/record=b2631352

Abstract

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 $\xi_0$ in $S^2(V^*)$, the symmetric square of the dual of V we are concerned with the Orthogonal group $O(\xi_0)$, the subgroup of the General Linear Group $GL(V)$ that fixes $\xi_0$ and with invariants of this group.

We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of $O(\xi_0)$. Considering the $O(\xi_0)$ 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 $\xi_1, \xi_2, \dots$ are be generated from $\xi_0$ by applying the action of the Steenrod Algebra to $S^2(V^*)$ 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 $O(\xi_0)$ when the dimension of the vector space is $4$ the finite field is $\mathbb{F}_3$ and the quadratic form is $\xi_0=x_1^2+x_2^2+x_3^2+x_4^2$ and we are able to establish an explicit presentation of $O(\xi_0)$ in this case.

Item Type: Thesis (PhD) Doctoral Invariants, Orthogonal group, Finite fields. Q Science > QA Mathematics College of Science and Engineering > School of Mathematics and Statistics > Mathematics Kropholler, Prof Peter 2008 Dr Sue Barnes glathesis:2008-300 Copyright of this thesis is held by the author. 19 Jun 2008 10 Dec 2012 13:17 https://theses.gla.ac.uk/id/eprint/300