Mirkovi\'c-Vilonen basis in type A_1

TL;DR

This paper explores the Mirković-Vilonen basis in type A_1, demonstrating its equivalence to the dual canonical basis for SL_2(C) at q=1 through geometric Satake correspondence.

Contribution

It establishes the coincidence of the Mirković-Vilonen basis with the dual canonical basis in the specific case of SL_2(C).

Findings

Mirković-Vilonen basis coincides with dual canonical basis at q=1 for SL_2(C)

Uses geometric Satake equivalence to relate basis constructions

Provides explicit basis identification in type A_1

Abstract

Let G be a reductive connected algebraic group over the field of complex numbers. Through the geometric Satake equivalence, the fundamental classes of the Mirkovi\'c-Vilonen cycles define a basis in each tensor product of rational irreducible representations of G. In the case G=SL_2(C), we show that this basis coincides with the dual canonical basis at q=1.