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Bin LI; Hechun ZHANG

doi:10.1007/s11464-010-0073-x

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Front. Math. China ›› 2011, Vol. 6 ›› Issue (4) : 689-706. DOI: 10.1007/s11464-010-0073-x

RESEARCH ARTICLE

Path realization of crystal B(∞)

Bin LI ,
Hechun ZHANG

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Abstract

A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equivalence classes of paths. With these induced operators, an explicit realization of $B (∞)$ is given in terms of equivalence classes of paths, where $B (∞)$ is the crystal base of the negative part of a quantum group $U q (g)$ . Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when $g$ is of finite type.

Keywords

Path / crystal / root operator

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Bin LI, Hechun ZHANG. Path realization of crystal

B (∞)

. Front Math Chin, 2011, 6(4): 689‒706 https://doi.org/10.1007/s11464-010-0073-x

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