Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 397-415, 2017

# Polytopes, quasi-minuscule representations and rational surfaces

## Jae-Hyouk Lee, Mang Xu, Jiajin Zhang

#### Received December 14, 2015.   First published May 4, 2017.

Jae-Hyouk Lee, Department of Mathematics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, 120-750 Seoul, Korea, e-mail: jaehyoukl@ewha.ac.kr; Mang Xu, Department of Mathematics, Southwest Jiaotong University, 2nd Ring Rd., 610031 Chengdu, Sichuan, P. R. China, e-mail: xumang@home.swjtu.edu.cn; Jiajin Zhang, Department of Mathematics, Sichuan University, Renmin South Rd 3rd Section, XiaoTianZhu, Wuhou Qu, 610065 Chengdu, Sichuan, P. R. China, e-mail: jjzhang@scu.edu.cn

Abstract: We describe the relation between quasi-minuscule representations, polytopes and Weyl group orbits in Picard lattices of rational surfaces. As an application, to each quasi-minuscule representation we attach a class of rational surfaces, and realize such a representation as an associated vector bundle of a principal bundle over these surfaces. Moreover, any quasi-minuscule representation can be defined by rational curves, or their disjoint unions in a rational surface, satisfying certain natural numerical conditions.

Keywords: rational surface; minuscule representation; polytope

Classification (MSC 2010): 14J26, 14N20

DOI: 10.21136/CMJ.2017.0676-15

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