Coordinate rings and birational charts
Sergey Fomin, George Lusztig

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