PBW Degenerate Schubert Varieties: Cartan Components and Counterexamples
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PBW Degenerate Schubert Varieties: Cartan Components and Counterexamples Igor Makhlin1,2 Received: 10 April 2019 / Accepted: 3 December 2019 / © Springer Nature B.V. 2019
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 sl6 . The counterexamples are constructed with the help of a study of the Cartan components appearing in this context. Keywords Semisimple Lie algebras · PBW degenerations · Demazure modules · Schubert varieties Mathematics Subject Classification (2010) 14M15 · 17B10 · 05E10
1 Introduction Over the last decade PBW degenerations of representations and flag varieties proved to be a diverse and fruitful research topic ([1, 4, 5, 7–9] and many others). Let us briefly outline the definitions of these objects. Consider a semisimple Lie algebra g and the subalgebra n− ⊂ g spanned by negative root vectors, an integral dominant g-weight λ and the irreducible representation Lλ . The PBW filtration on U (n− ) defines a filtration on Lλ via action on the highest weight vector, the Presented by: Peter Littelmann Igor Makhlin
[email protected] 1
Skolkovo Institute of Science and Technology, Center for Advanced Studies, Bolshoy Boulevard 30, bld. 1, Moscow, 121205, Russia
2
National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, Ulitsa Usacheva 6, Moscow, 119048, Russia
I. Makhlin
associated graded space Laλ is the PBW degenerate representation (of the abelian Lie algebra na− ). The space Laλ is then seen to be acted upon by the abelian Lie group N−a = Cdim n− , the closure Fλa of the N−a -orbit of the highest weight point in P(Laλ ) is the corresponding PBW degenerate flag variety. Let us restrict our attention to algebras g of type A. Here, a fundamental property of the degenerate flag varieties which makes them especially interesting is that they depend on the highest weight λ the same way as flag varieties do themselves. Let λ = a1 ω1 + . . . + an−1 ωn−1 for fundamental weights ωi , then Fλa is determined by the Weyl group stabilizer of λ, i.e. the set of such i that ai > 0. This was first proved in [7] by constructing a degenerate Pl¨ucker embedding of sorts for Fλa . A highly important fact, from which the Pl¨ucker embedding is obtained, is that Laλ can be realized as the cyclic submodule generated by the highest weight ve
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