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Generic semistability for reductive group actions

Authors: Dao Phuong Bac and Donghoon Hyeon
Journal: Proc. Amer. Math. Soc. 144 (2016), 4115-4124
MSC (2010): Primary 14L24
Published electronically: May 6, 2016
MathSciNet review: 3531165
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Abstract: We prove that given a reductive algebraic group $ G$ and a rational representation $ \rho : G \to \mathrm {GL}(V)$ defined over an algebraically closed field of characteristic 0, $ v \in V$ is generically semistable, i.e., $ 0 \not \in \overline {T.v}$ for a general maximal torus $ T$ if and only if $ v$ is semistable with respect to the induced action of the center of $ G$. The proof is obtained through a detailed description of the relation between the state polytope with respect to the maximal torus $ T$ of $ G$ and the state polytope with respect to $ T \cap [G,G]$. We also consider the case of solvable groups and prove that the generic semistability implies the center semistability but not the other way around.

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Dao Phuong Bac
Affiliation: Center for Geometry and its Applications, Department of Mathematics, Pohang University of Science and Technology, Pohang, Gyungbuk, Republic of Korea – and – Department of Mathematics, VNU University of Science, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

Donghoon Hyeon
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea

Keywords: Geometric Invariant Theory, state polytope
Received by editor(s): September 14, 2015
Received by editor(s) in revised form: December 2, 2015
Published electronically: May 6, 2016
Additional Notes: The first-named author was supported in part by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2014.34. The second-named author was supported by the SNU Invitation Program for Distinguished Scholar, the Research Resettlement Fund for the new faculty of Seoul National University, and the following grants funded by the government of Korea: NRF grant 2011-0030044 (SRC-GAIA) and NRF grant NRF-2013R1A1A2010649.
Communicated by: Lev Borisov
Article copyright: © Copyright 2016 American Mathematical Society

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