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 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.
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Acknowledgments
The author would like to thank Lara Bossinger, Xin Fang, Evgeny Feigin, Ghislain Fourier and Ievgen Makedonskyi for helpful discussions of these subjects. The work was partially supported by the grant RSF 19-11-00056. This research was supported in part by Young Russian Mathematics award.
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Presented by: Peter Littelmann
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Makhlin, I. PBW Degenerate Schubert Varieties: Cartan Components and Counterexamples. Algebr Represent Theor 23, 2315–2330 (2020). https://doi.org/10.1007/s10468-019-09943-y
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DOI: https://doi.org/10.1007/s10468-019-09943-y