Accessible Proof of Standard Monomial Basis for Coordinatization of Schubert Sets of Flags

TL;DR

This paper provides an accessible proof for the standard monomial basis of Schubert varieties in flag manifolds, using scanning tableaux to simplify the understanding of their algebraic structure.

Contribution

It introduces a more direct proof of the standard monomial basis for Schubert varieties using scanning tableaux, simplifying previous complex methods.

Findings

Provides a new proof of the basis using scanning tableaux

Shows the basis is a weight basis for Demazure modules

Expresses key polynomials as sums over tableaux weights

Abstract

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GL(n). This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases.