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 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.

近年来，在几篇论文中定义并研究了Demazure模块和Schubert变种的PBW退化。在A型中获得了各种有趣的特性（例如，这些PBW退化自然嵌入到相应的退化表示和标志变体中），但仅对Weyl基元素或最高重量有所限制。我们显示由于以下定义问题，这些属性不能完全通用。简并品系取决于用于定义它的最高权重，而不仅取决于其Weyl基团稳定剂（PBW简并旗品以及经典舒伯特品系就是这种情况）。也许令人惊讶的是，最小的反例仅针对\（\ mathfrak {sl} _ {6} \）出现。通过研究在这种情况下出现的Cartan组件来构造反例。