Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2014; 51(2): 427-442

Printed March 1, 2014

https://doi.org/10.4134/JKMS.2014.51.2.427

Copyright © The Korean Mathematical Society.

Crystal $B(\lambda)$ in $B(\infty)$ for $G_2$ type Lie algebra

Min Kyu Kim and Hyeonmi Lee

Gyeongin National University of Education, Hanyang University

Abstract

A previous work gave a combinatorial description of the crystal $B(\infty)$, in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present an explicit description of the crystal~$B(\lambda)$, in terms of the marginally large tableaux, for the $G_2$ Lie algebra type. We also provide a new description of $B(\la)$, in terms of Nakajima monomials, that is in natural correspondence with our tableau description.

Keywords: crystal base, $G_2$ type Lie algebra, marginally large tableau, Nakajima monomial

MSC numbers: 17B37, 81R50

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