Richardson varieties, projected Richardson varieties and positroid varieties

TL;DR

This survey reviews Richardson and positroid varieties, their combinatorics, parametrizations, and algebraic properties, highlighting recent results and connections to total positivity and plabic graphs.

Contribution

It provides an overview of known results on Richardson and positroid varieties, with some original insights into their algebraic structure and decompositions.

Findings

Coordinate rings of open Richardson varieties are UFDs

Deodhar decomposition is not a stratification in Lie type A

Explicit descriptions of Deodhar decomposition via submatrix ranks

Abstract

This is a survey article on Richardson varieties and their combinatorics. A Richardson variety is the intersection, inside the flag manifold GL_n/B_+, of a Schubert cell (B_- u B_+)/B_+ and an opposite Schubert cell (B_+ w B_+)/B_+ (or the similar intersection of Schubert varieties). In this survey, we provide an overview of what is known about (1) homogeneous coordinate rings of Richardson varieties, their bases and degenerations (2) parametrizations of Richardson varieties using Bott-Samelson varieties (3) Deodhar's decompositions of the flag manifold and of Richardson varieties within it and (4) total positivity in the flag manifold. We also provide an overview of the combinatorics of positroid varieties, their relations to Richardson varieties, and how they are parametrized using plabic graphs. Most of this survey is an overview of other authors' work over the last forty years, but there are also some minor original results: For example, that coordinate rings of open Richardson varieties are UFD's (Corollary 3.23), that the Deodhar decomposition is not a stratification in Lie type A (Section 4.3) and explicit descriptions of the Deodhar decomposition in terms of ranks of submatrices (Section 4.4).