Path realization of crystal B(\infty )

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(\infty ) is given in terms of equivalence classes of paths, where B(\infty ) is the crystal base of the negative part of a quantum group U_q(\mathfrak{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 \mathfrak{g} is of finite type.

Key words： Path; crystal; root operator

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