Seshadri stratification for Schubert varieties and Standard Monomial Theory

Abstract

Seshadri stratifications have been defined by the authors to generalize the Lakshmibai-Seshadri paths (LS-paths) from Schubert varieties to projective varieties with such a stratification. We investigate the Seshadri stratification on a Schubert variety arising from its Schubert subvarieties.

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

“…We show that the set of LS-paths LS + (λ) coincides with the fan of monoids Γ of V and thus give an algebro-geometric interpretation of the LS-path model of a representation. In addition we show that the standard monomials defined in [28] are indeed representatives of the non-zero leaves of the quasi-valuation V for this Seshadri stratification, and the notion of the standardness of a product in [11] and in [28] coincide. Also, the stratification is normal and balanced in this case.…”

“…Now fix a Schubert variety X(τ ), τ ∈ W/W P , and consider its embedding in P(V (λ) τ ) where V (λ) τ ⊆ V (λ) is the Demazure module of τ . In [11] we have proved that the collection of Schubert subvarieties contained in X(τ ) and of the functions of extremal weight in V (λ) * τ defines a Seshadri stratification for X(τ ). Moreover, the main point in [11] is to show that the standard monomial theory defined in [28] fits into the concept of our theory.…”

“…In [11] we have proved that the collection of Schubert subvarieties contained in X(τ ) and of the functions of extremal weight in V (λ) * τ defines a Seshadri stratification for X(τ ). Moreover, the main point in [11] is to show that the standard monomial theory defined in [28] fits into the concept of our theory. More precisely, let V be the quasivaluation arising from this Seshadri stratification.…”

“…So we have a collection of subvarieties X(σ) of X(τ ), indexed by the partially ordered set A τ such that κ ≤ σ if an only if X(κ) ⊆ X(σ). In addition, all the subvarieties are smooth in codimension one by [11,Corollary 3.3]. The covering relations correspond to codimension one subvarieties since the length function gives the dimension of a Schubert variety and this function coincides with the length in Definition 2.3.…”

“…The same construction, slightly modified, holds for any symmetrizable Kac-Moody group. For the general case, all proofs and further details can be found in [11].…”