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.Comment: 18 page
