Vector invariants for two-dimensional orthogonal groups over finite fields

Abstract

Let $\mathbb {F}_{q}$ be a finite field of characteristic 2 and $O_{2}^{+}(\mathbb {F}_{q})$ be the 2-dimensional orthogonal group over $\mathbb {F}_{q}$. Consider the standard representation V of $O_{2}^{+}(\mathbb {F}_{q})$ and the ring of vector invariants $\mathbb {F}_{q}[mV]^{O_{2}^{+}(\mathbb {F}_{q})}$ for any $m\in \mathbb {N}^{+}$. We prove a first main theorem for $(O_{2}^{+}(\mathbb {F}_{q}),V)$, i.e., we find a minimal generating set for $\mathbb {F}_{q}[mV]^{O_{2}^{+}(\mathbb {F}_{q})}$. As a consequence, we derive the Noether number $\upbeta _{mV}(O_{2}^{+}(\mathbb {F}_{q}))=\mathrm{max}\{q-1,m\}$. We construct a free basis for $\mathbb {F}_{q}[2V]^{O_{2}^{+}(\mathbb {F}_{q})}$ over a suitably chosen homogeneous system of parameters. We also obtain a generating set for the Hilbert ideal of $\mathbb {F}_{q}[mV]^{O_{2}^{+}(\mathbb {F}_{q})}$ which shows that the Hilbert ideal can be generated by invariants of degree $\leqslant q-1=\frac{|O_{2}^{+}(\mathbb {F}_{q})|}{2}$, confirming the conjecture of Derksen and Kemper (Computational invariant theory. Encyclopaedia of mathematical sciences, vol 130. Springer, New York, 2002, Conjecture 3.8.6 (b)) in this particular case.