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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Abstract
Let $V$ be a nonzero finite dimensional vector space over a finite field $\mathbb{F}_q$ of odd characteristic.
Fixing a nonsingular 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.
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