Adapted algebras and standard monomials

Abstract

Let G be a complex semisimple Lie group. The aim of this article is to compare two basis for G-modules, namely the standard monomial basis and the dual canonical basis. In particular, we give a su cient condition for a standard monomial to be an element of the dual canonical basis and vice versa.Let G denote a semisimple, simply connected, algebraic group deÿned over an algebraically closed ÿeld k of arbitrary characteristic. We ÿx a Borel subgroup B and a maximal torus T ⊂ B, denote by W the Weyl group of G with respect to T . For a dominant weight let V ( ) be the corresponding Weyl module and let Q be the parabolic subgroup which normalizes a highest weight vector. Let L be the corresponding ample line bundle on G=Q. Consider the embedding G=Q ,→ P(V ( )) given by the global sections H 0 (G=Q; L ) V ( ) * .The aim of standard monomial theory is to give a presentation of the ring R = n¿0 H 0 (G=Q; L n ), which is compatible with the natural subvarieties of G=Q as

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

“…the compatibility of the standard monomial theory to Schubert sub-varieties of X(τ ); (2). the surjectivity of the restriction maps for line bundles to these sub-varieties; (3). surjectivity of the multiplication maps for line bundles; (4).…”

“…In this context: it would be interesting to know whether other known bases of V (λ) * τ are compatible with the filtration and can be used as representatives x a for the leaves associated to an LS-path of degree 1, and hence serve as starting point for a standard monomial theory in the sense of Theorem 5.1. It has been shown in the finite type case in [3] that the elements of the path basis satisfying certain compatibility conditions with respect to a reduced decomposition of the longest element in the Weyl group belong to (up to multiplication by non-zero scalars) the dual canonical basis, so the dual canonical basis of V (λ) * τ is a good candidate. he properties of the Mirković-Vilonen (MV) basis and its dual basis proved in [2], Section 5.7, suggest that the elements of the dual basis could be compatible with the filtration, and hence serve as representatives for the leaves as well.…”