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

Abstract

We consider a desingularization Γ of a Richardson variety in the variety of complete flags, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. For any projective embedding of Z via a complete linear system, we construct a basis of the homogeneous coordinate ring of Γ inside Z, indexed by combinatorial objects which we call w 0 -standard tableaux. and we have w max (i(J 6,1 )) = s 1 s 3 s 2 s 1 s 3 = [4231] = w 0 , hence T is not w 0 -standard.As an immediate consequence of Theorems 3.3 and 4.20, we have Corollary 4.22. Assume that m is regular, so we embed Γ i in the projective space P(H 0 (Γ i , L i,m ) * ). A basis of the homogeneous coordinate ring of Γ i is given by the w 0 -standard monomials of shape (i, dm) for all d ∈ N.

Effective Divisors on Bott–samelson Varieties

“…For the case w = w • -that is, when X(u)∩w·X(v) is a Richardson variety-a similar construction was given by Balan [2] (following a suggestion of Brion), and the geometry of the fiber ϕ −1 (w • ) was studied by Escobar [8].…”

“…Now restrict to the basic diagram (2), and consider the dense open set To see that ϕ −1 X(ws β ) is reduced along Σ, again restrict to the basic diagram (2). Recall that µ • : X(w) • × E → X(w) • ∪ X(ws β ) • is given by µ • (uwB, wpB) = uwpB, and by the diagram (3), it is identified with the projection of a trivial A 1 -bundle.…”

“…In the infinite-dimensional case, however, Richardson varieties usually cannot be written as intersections of translated Schubert varieties in this way. 2 Suppose α = (α 1 , . .…”

“…, γ d by γ i = s d s d−1 · · · s i+1 (α i ), and write r i = ρ, γ ∨ i , where ρ is the sum of the fundamental weights. If the word α is reduced, then all the roots γ i are positive and r i ≥ 1 for each 2 Ed Richmond has pointed out that in affine type A1, given elements x, y ∈ W , one can find u, v ∈ W such that X y x = X(u) ∩ w · X(v) if and only if dim X y x = ℓ(x) − ℓ(y) = 1.…”

The Grassmannian Variety

“…In [79], the author proves the vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties. A standard monomial theory has been developed for Bott-Samelson varieties in [47]; see also [2]. In [25], the authors study certain toric varieties associated to Bott-Samelson varieties.…”