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.
One of important open problems is to give a completely geometric construction of the theory.
Seshadri (1978) 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.
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.
Lakshmibai (2003) and Musili (2003) and Seshadri (2012) give detailed descriptions of the early development of standard monomial theory.