Parametrizations of canonical bases and irreducible components of nilpotent varieties

Details Parametrizations of canonical bases and irreducible components of nilpotent varieties Parametrizations of canonical bases and irreducible components of nilpotent varieties

2011-10-13

arxiv.org/abs/1110.2937v1

Mathematics

Quantum Algebra

23 pages

Scientific paper

It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis. For each reduced expression of a Weyl group element, Gei{\ss}, Leclerc and Schr\"oer has recently given a parametrization of a subset of irreducible components in studying the cluster structure of the coordinate ring of the corresponding unipotent subgroup. In this paper we show that their parametrization is compatible with Lusztig's parametrization of canonical basis. We also give some interpretations of Lusztig's transition map in terms of irreducible components of nilpotent varieties.

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