The matrix between PBW basis and semicanonical basis of $U^+(sl_n(\mathbb{C}))$

TL;DR

This paper proves that the transition matrix between a specific PBW basis and the semicanonical basis of the positive part of the quantum group for sl_n(C) is upper triangular unipotent under compatible orders, revealing a structured relationship.

Contribution

It establishes the upper triangular unipotent nature of the matrix between PBW and semicanonical bases for U^+(sl_n(C)) under compatible orders, a new structural insight.

Findings

Matrix is upper triangular unipotent

Order compatibility with partial order $\, ext{deg}$

Structural relationship between bases

Abstract

In this paper, we prove that the matrix between a special PBW basis and the semicanonical basis of $U^+(sl_n(\mathbb{C}))$ is upper triangular unipotent under any order which is compatible with the partial order $\leq_{deg}$.