Seshadri stratifications and Schubert varieties: a geometric construction of a standard monomial theory

Abstract

A standard monomial theory for Schubert varieties is constructed exploiting (1) the geometry of the Seshadri stratifications of Schubert varieties by their Schubert subvarieties and (2) the combinatorial LS-path character formula for Demazure modules. The general theory of Seshadri stratifications is improved by using arbitrary linearization of the partial order and by weakening the definition of balanced stratification.

On normal Seshadri stratifications

“…First of all the bonds b p,q , p covering q in A, fullfill the gcd conditions on chains as is proved in [9,Corollary 4.8] (the proof of Corollary 4.8 in [9] is for the application to Schubert varieties, however the same proof holds for arbitrary stratification of LS-type). So (1) is proved.…”

“…It is not hard to see that there are only 14 of them (see also [14, Table 1]) with a = (1, 1, 1, 1), (1, 1, 1, 3), (1, 1, 2, 2), (1, 1, 2, 4), (1,1,4,6), (1,2,2,5), (1,2,3,6), (1,2,6,9), (1,3,4,4), (1,3,8,12), (1,4,5,10), (1,6,14,21), (2,3,3,4), (2,3,10,15).…”

“…As an application, the Lakshmibai-Seshadri path model [20,21] for a Schubert variety is recovered from the Seshadri stratification consisting of Schubert subvarieties contained in it (see [8], [9] for details).…”