Symplectic PBW Degenerate Flag Varieties; PBW Tableaux and Defining Equations

TL;DR

This paper introduces PBW-semistandard tableaux for symplectic Lie algebra modules, establishing their role in parametrizing bases of coordinate rings and deriving explicit defining relations for PBW degenerate flag varieties.

Contribution

It defines new PBW-semistandard tableaux, links them to monomials in the polytope, and explicitly describes the ideal generators for PBW degenerate symplectic flag varieties.

Findings

PBW-semistandard tableaux biject with monomials in the polytope

They parametrize bases of coordinate rings of flag varieties

Explicit degenerate relations generate the defining ideal

Abstract

We define a set of PBW-semistandard tableaux that is in a weight preserving bijection with the set of monomials corresponding to integral points in the Feigin-Fourier-Littelmann-Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the homogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety. These relations consist of type A degenerate Pl\"ucker relations and a set of degenerate linear relations that we obtain from De Concini's linear relations.