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A diagram algebra for Soergel modules corresponding to smooth Schubert varieties


Author: Antonio Sartori
Journal: Trans. Amer. Math. Soc. 368 (2016), 889-938
MSC (2010): Primary 16W50; Secondary 13F20, 05E05, 17B10
DOI: https://doi.org/10.1090/tran/6346
Published electronically: May 11, 2015
MathSciNet review: 3430353
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Abstract: Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category $ \mathcal {O}(\mathfrak{gl}_n)$ which show up in categorification of $ \mathfrak{gl}(1\vert 1)$-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras.


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Additional Information

Antonio Sartori
Affiliation: Mathematisches Institut, Endenicher Allee 60, Universität Bonn, 53115 Bonn, Germany
Address at time of publication: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg im Breisgau, Germany
Email: antonio.sartori@math.uni-freiburg.de

DOI: https://doi.org/10.1090/tran/6346
Keywords: Diagram algebra, symmetric polynomials, category $\mathcal{O}$, Soergel modules, Khovanov algebra
Received by editor(s): October 14, 2013
Received by editor(s) in revised form: December 4, 2013
Published electronically: May 11, 2015
Additional Notes: This work has been supported by the Graduiertenkolleg 1150, funded by the Deutsche Forschungsgemeinschaft.
Article copyright: © Copyright 2015 American Mathematical Society