Degenerate Schubert Varieties in Type A

TL;DR

This paper introduces rectangular elements in the symmetric group and demonstrates that their associated degenerate Schubert varieties in type A are actually Schubert varieties in larger partial flag varieties, extending previous PBW degeneration results.

Contribution

It establishes a new connection between degenerate Schubert varieties and classical Schubert varieties in larger flag varieties for rectangular elements in type A.

Findings

Degenerate Schubert varieties correspond to larger flag varieties.

Degenerate Demazure modules are isomorphic to classical Demazure modules.

Generalizes previous PBW degeneration results.

Abstract

We introduce rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated to a rectangular element is indeed a Schubert variety in a partial flag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated to a rectangular element is isomorphic to the Demazure module for this particular Schubert variety of larger rank. This generalizes previous results by Cerulli Irelli, Lanini and Littelmann for the PBW degenerate flag variety.