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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 anincreasing sequence of subspaces with those dimensions. The set of all suchflags (the flag manifold) can be projectively coordinatized using products ofminors of a matrix. These products are indexed by tableaux on a Young diagram.A basis of "standard monomials" for the vector space generated by suchprojective coordinates over the entire flag manifold has long been known. ASchubert variety is a subset of flags specified by a permutation. Lakshmibai,Musili, and Seshadri gave a standard monomial basis for the smaller vectorspace generated by the projective coordinates restricted to a Schubert variety.Reiner and Shimozono made this theory more explicit by giving a straighteningalgorithm for the products of the minors in terms of the right key of a Youngtableau. Since then, Willis introduced scanning tableaux as a more direct wayto obtain right keys. This paper uses scanning tableaux to give more-directproofs 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 aDemazure module for a Borel subgroup of GL(n). This paper contains a completeproof that the characters of these modules (the key polynomials) can beexpressed as the sums of the weights for the tableaux used to index thestandard monomial bases.

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