Standard Monomial Theory of RR Varieties

Abstract

We construct the RR varieties as the fiber products of Bott-Samelson varieties over Richardson varieties. We study their homogeneous coordinate rings and standard monomial theory.

Standard monomial theory for desingularized Richardson varieties in the flag variety GL(n)/B

“…This shows that the w 0 -standard monomials span H 0 (Γ i , L i,m ). Note that alternate bases of H 0 (Z i , L i,m ) for certain pairs (i, m) have been constructed in [39], and the fibred products Z × GL(n,C)/B Z ′ have been studied from this point of view in [11].Sections are organized as follows: in Section 1, we first fix notation and recall information on Bott-Samelson varieties Z i , and then show that the fibre Γ i of Z i → G/B over w 0 B is a desingularization of the Richardson variety X v w ; this fact is most certainly known to experts, but has not, to our knowledge, appeared in the literature. In Section 2, we recall the main results about SMT for Bott-Samelson varieties from [28], in particular the definition of standard tableaux.…”

“…This shows that the w 0 -standard monomials span H 0 (Γ i , L i,m ). Note that alternate bases of H 0 (Z i , L i,m ) for certain pairs (i, m) have been constructed in [39], and the fibred products Z × GL(n,C)/B Z ′ have been studied from this point of view in [11].…”