# PBW degenerate Schubert varieties: Cartan components and counterexamples

## TL;DR

This paper investigates PBW degenerations of Schubert varieties, revealing limitations in their properties and providing counterexamples, especially in the case of rak{sl}_6, through analysis of Cartan components.

## Contribution

It demonstrates that certain properties of PBW degenerations do not hold universally and introduces counterexamples based on Cartan component analysis.

## Key findings

- Counterexamples in rak{sl}_6 show limitations of PBW degenerations.
- Properties depend on highest weight, not just Weyl group stabilizer.
- Counterexamples challenge previous assumptions about PBW degenerations.

## Abstract

In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type $\mathrm A$ but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for $\mathfrak{sl}_6$. The counterexamples are constructed with the help of a study of the Cartan components appearing in this context.