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

DOI:
https://doi.org/10.1090/proc/13110

Published electronically:
May 6, 2016

MathSciNet review:
3531165

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that given a reductive algebraic group and a rational representation defined over an algebraically closed field of characteristic 0, is generically semistable, i.e., for a *general* maximal torus if and only if is semistable with respect to the induced action of the center of . The proof is obtained through a detailed description of the relation between the state polytope with respect to the maximal torus of and the state polytope with respect to . 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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Additional Information

**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

Email:
bacdp@postech.ac.kr, bacdp@vnu.edu.vn, dpbac.vnu@gmail.com

**Donghoon Hyeon**

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

Email:
dhyeon@snu.ac.kr

DOI:
https://doi.org/10.1090/proc/13110

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