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Polynomial Invariants of Orthogonal Groups

By 蕭衡慶 and Heng-ching Hsiao


[[abstract]]Let Q_{n}^{-}=x_{1}^{2}-x_{2}^{2}+...+x_{n-1}^{2}-εx_{n}^{2} be a nondegenerate quadratic form over the finite field Fq with charFq is not equal to 2 where ε is a non-square in Fq, and let O(Fq^{n},Q_{n}^{-}) be the associated orthogonal group. Let O(Fq^{n},Q_{n}^{-}) act linearly on the polynomial ring Fq[x_{1},...,x_{n}]. In this thesis we try to find the invariant subring Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})} with explicit generators. In fact, we find the invariant elements in Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})}, and denote the subring which these invariant elements generate by R_{n}^{*}. We show that Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})} is integral over R_{n}^{*}, and show that Fq[x_{1},...,x_{n}]^{O(Fq^{n},Q_{n}^{-})} and R_{n}^{*} have the same quotient field. As n=2,4, we show that R_{n}^{*} is a UFD, and hence is integrally closed. Thus Fq[x1,x2]^{O(Fq^{2},Q_{2}^{-})}=R_{2}^{*} and Fq[x1,x2,x3,x4]]^{O(Fq^{4},Q_{4}^{-})}=R_{4}^{*}, so we get the generators of invariant subring when n=2,4.

Topics: 二次型;正交群;不變子環, quadratic form;orthogonal group;invariant subring, [[classification]]37
Year: 2011
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