Standard Monomial Theory for desingularized Richardson varieties in the flag variety GL(n)/B

TL;DR

This paper constructs a basis for the coordinate ring of a desingularized Richardson variety in the flag variety using combinatorial objects called w_0-standard tableaux, advancing understanding of their algebraic structure.

Contribution

It introduces a new combinatorial basis for the coordinate ring of desingularized Richardson varieties in GL(n)/B.

Findings

Basis indexed by w_0-standard tableaux constructed

Provides explicit combinatorial description of the coordinate ring

Advances algebraic understanding of Richardson varieties

Abstract

We consider a desingularization Gamma of a Richardson variety X_w^v=X_w \cap X^v in the flag variety Fl(n)=GL(n)/B, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous coordinate ring of Gamma inside Z, indexed by combinatorial objects which we call w_0-standard tableaux.