Sábháilte in:
| Príomhchruthaitheoir: | |
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| Formáid: | Preprint |
| Foilsithe / Cruthaithe: |
2016
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| Ábhair: | |
| Rochtain ar líne: | https://arxiv.org/abs/1612.06039 |
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| _version_ | 1866908847633858560 |
|---|---|
| author | Chen, Yin |
| author_facet | Chen, Yin |
| contents | Let $\mathbb{F}_{q}$ be a finite field of characteristic $2$ and $O_2^+(\mathbb{F}_{q})$ be the $2$-dimensional orthogonal group of plus type 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 $β_{mV}(O_2^+(\mathbb{F}_{q}))=\max\{q-1,m\}$. 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 of the Hilbert ideal for $\mathbb{F}_{q}[mV]^{O_2^+(\mathbb{F}_{q})}$ which shows that the Hilbert ideal can be generated by invariants of degree $\leqslant q-1=\frac{|O_2^+(\mathbb{F}_{q})|}{2}$, positively confirming a conjecure of Derksen and Kemper for this particular case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1612_06039 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Vector invariants for two-dimensional orthogonal groups over finite fields Chen, Yin Commutative Algebra Let $\mathbb{F}_{q}$ be a finite field of characteristic $2$ and $O_2^+(\mathbb{F}_{q})$ be the $2$-dimensional orthogonal group of plus type 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 $β_{mV}(O_2^+(\mathbb{F}_{q}))=\max\{q-1,m\}$. 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 of the Hilbert ideal for $\mathbb{F}_{q}[mV]^{O_2^+(\mathbb{F}_{q})}$ which shows that the Hilbert ideal can be generated by invariants of degree $\leqslant q-1=\frac{|O_2^+(\mathbb{F}_{q})|}{2}$, positively confirming a conjecure of Derksen and Kemper for this particular case. |
| title | Vector invariants for two-dimensional orthogonal groups over finite fields |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/1612.06039 |