Parametrizations of canonical bases and irreducible components of nilpotent varieties

TL;DR

This paper demonstrates that the parametrization of irreducible components of nilpotent varieties by Geiß, Leclerc, and Schröer aligns with Lusztig's canonical basis parametrization, linking geometric and algebraic structures.

Contribution

It establishes the equivalence between two different parametrizations of canonical bases and irreducible components, bridging geometric and algebraic approaches in representation theory.

Findings

Geiß-Leclerc-Schröer's parametrization matches Lusztig's canonical basis.

Provides a geometric realization of the crystal basis for quantum groups.

Unifies different parametrization methods in the study of nilpotent varieties.

Abstract

It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Gei{\ss}, Leclerc and Schr\"{o}er has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.