Frontiers of Mathematics in China >
Path realization of crystal
Received date: 26 Feb 2010
Accepted date: 17 Jun 2010
Published date: 01 Aug 2011
Copyright
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 is given in terms of equivalence classes of paths, where is the crystal base of the negative part of a quantum group . 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 is of finite type.
Key words: Path; crystal; root operator
Bin LI , Hechun ZHANG . Path realization of crystal [J]. Frontiers of Mathematics in China, 2011 , 6(4) : 689 -706 . DOI: 10.1007/s11464-010-0073-x
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