Seshadri stratification for Schubert varieties and standard monomial theory

The theory of Seshadri stratifications has been developed by the authors with the intention to build up a new geometric approach towards a standard monomial theory for embedded projective varieties with certain nice properties. In this article, we investigate the Seshadri stratification on a Schubert variety arising from its Schubert subvarieties. We show that the standard monomial theory developed in Littelmann ( J. Am. Math. Soc. 11 ( 3 ) (1998) 551–567) is compatible with this new strategy.