Standard monomial theory

In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of the results have been extended to Kac–Moody algebras and their groups.
There are monographs on standard monomial theory by and and survey articles by and 
One of important open problems is to give a completely geometric construction of the theory.
History.
 introduced monomials associated to standard Young tableaux.
 (see also ) used Young's monomials, which he called standard power products, named after standard tableaux, to give a basis for the homogeneous coordinate rings of complex Grassmannians. initiated a program, called standard monomial theory, to extend Hodge's work to varieties "G"/"P", for "P" any parabolic subgroup of any reductive algebraic group in any characteristic, by giving explicit bases using standard monomials for sections of line bundles over these varieties. The case of Grassmannians studied by Hodge corresponds to the case when "G" is a special linear group in characteristic 0 and "P" is a maximal parabolic subgroup. Seshadri was soon joined in this effort by V. Lakshmibai and Chitikila Musili. They worked out standard monomial theory first for minuscule representations of "G" and then for groups "G" of classical type, and formulated several conjectures describing it for more general cases. proved their conjectures using the Littelmann path model, in particular giving a uniform description of standard monomials for all reductive groups.
 and and give detailed descriptions of the early development of standard monomial theory.