Coordinate rings and birational charts

TL;DR

This paper describes the coordinate rings of a semisimple algebraic group and its related spaces using birational charts linked to total positivity, providing explicit algebraic descriptions.

Contribution

It introduces explicit descriptions of coordinate rings of G, U, and G/U using specific birational charts related to total positivity.

Findings

Coordinate rings expressed via birational charts

Connections established with total positivity

Explicit algebraic descriptions provided

Abstract

Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational charts introduced in [L94, L19] in connection with the study of total positivity.