Combinatorial Structures in the Coordinate Rings of Schubert Varieties

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. This dissertation uses the recently introduced notion of scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. This basis is a weight basis for the dual of a Demazure module for a Borel subgroup of the general linear group.;The most famous of the above flag manifolds are the Grassmann manifolds of flags that consist of a single subspace. The Plucker relations which define the Grassmann manifolds as projective varieties are well known. Grassmann manifolds are examples of minuscule flag manifolds. We study the generalized Plucker relations for minuscule flag manifolds independent of Lie type. To do this we combinatorially model the Plucker coordinates based on Wildberger's construction of minuscule Lie algebra representations; it uses the colored partially ordered sets known as minuscule posets. We obtain, uniformly across Lie type, descriptions of the Plucker relations of "extreme weight". We show that these are "supported" by "double-tailed diamond" sublattices of minuscule lattices. From this, we obtain a complete set of Plucker relations for the exceptional minuscule flag manifolds. These Plucker relations are straightening laws for their coordinate rings.