Abstract
A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties. Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module. In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.
Highlights
The classical Borel-Weil theorem identifies the global sections Γ(G/B, Lλ) of a line bundle on a flag variety with the irreducible highest weight representation V (λ)
When the same line bundle is restricted to a Schubert variety Xw, the global sections Γ(Xw, Lλ) can be identified with the Demazure module Vw(λ)
We show in Theorem 15 that a graded piece of the homogeneous coordinate ring of a positroid variety can be identified with the cyclic Demazure module
Summary
The classical Borel-Weil theorem identifies the global sections Γ(G/B, Lλ) of a line bundle on a flag variety with the irreducible highest weight representation V (λ). When the same line bundle is restricted to a Schubert variety Xw, the global sections Γ(Xw, Lλ) can be identified with the Demazure module Vw(λ). Positroid varieties (see Section 4) are certain intersections of cyclically rotated Schubert varieties in the Grassmannian. They were introduced in Postnikov’s work [Pos] on the totally nonnegative Grassmannian, and subsequently studied in algebro-geometric terms by Knutson-Lam-Speyer [KLS13]. Our approach is based on the key observation (Theorem 1(iv)) that the dual canonical basis of the Grassmannian is invariant under signed cyclic rotation. This relies heavily on the work of Rhoades [Rho].
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