Kraśkiewicz-Pragacz modules and Ringel duality

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• 作者：Masaki Watanabe ^{mwata@ms.u-tokyo.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Schubert polynomials ; Kraśkiewicz&ndash ; Pragacz modules ; Highest weight categories ; Ringel duality ; B-modules
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：468
• 期：Complete
• 页码：1-23
• 全文大小：516 K

文摘

In  and  Kraśkiewicz and Pragacz introduced representations of the upper-triangular Lie algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302332&_mathId=si1.gif&_user=111111111&_pii=S0021869316302332&_rdoc=1&_issn=00218693&md5=d5fa10ac0221d8d55dc37b02250cf57d" title="Click to view the MathML source">bclass="mathContainer hidden">class="mathCode">$\mathfrak{b}$ whose characters are Schubert polynomials. In [12] the author studied the properties of Kraśkiewicz–Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual. This leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302332&_mathId=si1.gif&_user=111111111&_pii=S0021869316302332&_rdoc=1&_issn=00218693&md5=d5fa10ac0221d8d55dc37b02250cf57d" title="Click to view the MathML source">bclass="mathContainer hidden">class="mathCode">$\mathfrak{b}$-modules is compatible with Ringel duality functor.