Dual canonical bases for unipotent groups and base affine spaces

TL;DR

This paper establishes a parameterization of the dual canonical basis of the coordinate ring of unipotent groups using semi-standard Young tableaux, providing explicit formulas and applications to cluster variables.

Contribution

It introduces a novel parameterization of the dual canonical basis via Young tableaux and derives explicit formulas for basis elements.

Findings

Dual canonical basis parameterized by Young tableaux

Explicit formulas for basis elements

Application to cluster variables in $\

Abstract

Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\mathbb{C}[N]$ (and base affine spaces) can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis using the data of the corresponding semistandard Young tableau. We apply our results to study cluster variables in $\mathbb{C}[N]$.