Sums of squares and moment problems in equivariant situations
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- by Jaka Cimprič, Salma Kuhlmann and Claus Scheiderer PDF
- Trans. Amer. Math. Soc. 361 (2009), 735-765 Request permission
Abstract:
We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\mathbb {R}$ acting on an affine $\mathbb {R}$-variety $V$, we consider the induced dual action on the coordinate ring $\mathbb {R}[V]$ and on the linear dual space of $\mathbb {R}[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\mathbb R)$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\mathbb {R}[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\mathbb {R}[V]$ and associated quadratic modules of the (finitely generated) subring $\mathbb {R}[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\mathbb {R})$ is compact.References
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Additional Information
- Jaka Cimprič
- Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenija
- Email: cimpric@fmf.uni-lj.si
- Salma Kuhlmann
- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Room 142 McLean Hall, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
- MR Author ID: 293156
- Email: skuhlman@snoopy.usask.ca
- Claus Scheiderer
- Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, D-78457 Konstanz, Germany
- MR Author ID: 212893
- Email: claus.scheiderer@uni-konstanz.de
- Received by editor(s): November 19, 2006
- Published electronically: September 23, 2008
- Additional Notes: The third author was partially supported by the European RTNetwork RAAG, HPRN-CT-2001-00271
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 735-765
- MSC (2000): Primary 14P10, 14L30, 20G20
- DOI: https://doi.org/10.1090/S0002-9947-08-04588-1
- MathSciNet review: 2452823