PBW-degenerated Demazure modules and Schubert varieties for triangular elements

Abstract

We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sln+1. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.

Degenerate Schubert Varieties in Type A

“…As their constructions are similar to the twisted Schubert varieties, we call them degenerate Schubert varieties in degenerate flag varieties. One should compare this definition with the one given by the third author in [16].…”

“…Remark 3.9. The Kempf elements studied in [16], while being triangular, are not necessarily rectangular, for example s 1 s 2 s 3 s 2 is a Kempf element but it is not rectangular.…”

“…The first step of the proof is to show that the corresponding Demazure modules have the same dimension. The proof makes fully use of the basis constructed in [14,16] and its parametrisation by the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope, as well as all results we have developed for rectangular elements. Proof.…”

“…The concept of PBW degenerations can be applied to spherical varieties in general: one partially abelianizes the corresponding Lie algebra and algebraic group (see [14,10]) and considers its orbit in the associated graded module. A natural class of spherical varieties is that of Schubert varieties and a first step towards the analysis of PBW degenerate Schubert varieties has been carried out in [16], where for a triangular element in the Weyl group, a monomial basis of the homogeneous coordinate ring has been provided. From a combinatorial point of view, this basis can be identified with lattice points on a face of the previous polyhedron.…”

“…Rectangular elements. We define rectangular elements in a similar way to the definition of triangular elements in[16, Proposition 1].Definition 3.6. An element τ ∈ W is called rectangular if N(τ ) is a rectangular subset of Φ + .Example 3.7.…”