Degree bounds for separating invariants

If $V$ is a representation of a linear algebraic group $G$, a set $S$ of $G$-invariant regular functions on $V$ is called {\it separating\/} if the following holds: {\it If two elements $v,v'\in V$ can be separated by an invariant function, then there is an $f\in S$ such that $f(v)\neq f(v')$.} It is known that there always exist finite separating sets. Moreover, if the group $G$ is finite, then the invariant functions of degree $\leq |G|$ form a separating set. We show that for a non-finite linear algebraic group $G$ such an upper bound for the degrees of a separating set does not exist. If $G$ is finite, we define $\bsep(G)$ to be the minimal number $d$ such that for every $G$-module $V$ there is a separating set of degree $\leq d$. We show that for a subgroup $H \subset G$ we have $\bsep(H) \leq \bsep(G)\leq [G:H] \cdot\bsep(H)$, and that $\bsep(G)\leq \bsep(G/H) \cdot \bsep(H)$ in case $H$ is normal. Moreover, we calculate $\bsep(G)$ for some specific finite groups.

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Published 1 January 2010