Zero-separating invariants for linear algebraic groups

Article, Preprint English OPEN
Elmer, Jonathan ; Kohls, Martin (2014)
  • Publisher: Cambridge University Press
  • Related identifiers: doi: 10.1017/S0013091515000322
  • Subject: 13A50 | Mathematics - Commutative Algebra | Mathematics - Algebraic Geometry

Let $G$ be a linear algebraic group over a field $k$, and let $V$ be a $G$-module. Recall that the nullcone of $(G,V)$ is the set of points $v$ in $V$ with the property that $f(v)=0$ for every positive degree homogeneous invariant $f$ in $k[V]^G$. We define numbers $\delta(G,V)$ and $\sigma(G,V)$ associated with a given representation as follows: $\delta(G,V)$ is the smallest number $d$ such that, for any point $v$ in $V^G$ outside the nullcone, there exists an invariant $f$ of degree at most $d$ such that $f(v)$ is not zero; $\sigma(G,V)$ is the same thing with $V^G$ replaced by $V$. If k has positive characteristic, we show that $\delta(G,V)$ is infinite for all subgroups of $GL_2(k)$ containing a unipotent subgroup, and that $\sigma(G,V)$ is finite if and only if $G$ is finite. If $k$ has characteristic zero we show that $\delta(G,V)=1$ for all linear algebraic groups and that if $\sigma(G,V)$ is finite then the connected component of $G$ is unipotent.
  • References (12)
    12 references, page 1 of 2

    [1] Kalman Cziszter and Matyas Domokos. On the generalized Davenport constant and the Noether number. Cent. Eur. J. Math., 11(9):1605{1615, 2013.

    [2] Harm Derksen. Polynomial bounds for rings of invariants. Proc. Amer. Math. Soc., 129(4):955{963 (electronic), 2001.

    [3] Harm Derksen and Gregor Kemper. Computational invariant theory. Invariant Theory and Algebraic Transformation Groups, I. Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical Sciences, 130.

    [4] Jonathan Elmer and Martin Kohls. Separating invariants for the basic Ga-actions. Proc. Amer. Math. Soc., 140(1):135{146, 2012.

    [5] Jonathan Elmer and Martin Kohls. Zero-separating invariants for nite groups. To appear in J. of Algebra. Preprint available from http://arxiv.org/abs/1308.0991, 2013.

    [6] A. Fauntleroy. On Weitzenbock's theorem in positive characteristic. Proc. Amer. Math. Soc., 64(2):209{213, 1977.

    [7] Gene Freudenburg. Algebraic theory of locally nilpotent derivations, volume 136 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2006. Invariant Theory and Algebraic Transformation Groups, VII.

    [8] James E. Humphreys. Linear algebraic groups. Springer-Verlag, New York, 1975. Graduate Texts in Mathematics, No. 21.

    [9] Martin Kohls and Hanspeter Kraft. Degree bounds for separating invariants. Math. Res. Lett., 17(6):1171{1182, 2010.

    [10] Masayoshi Nagata. Invariants of a group in an a ne ring. J. Math. Kyoto Univ., 3:369{377, 1963/1964.

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