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arXiv · 2008.08864

Bigrassmannian permutations and Verma modules

Hankyung Ko·Volodymyr Mazorchuk·Rafael Mrđen
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Abstract

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type $A$. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $w\in S_n$ into the dominant Verma module are shown to be determined by the essential set of $w$ and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

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Hankyung Ko, Volodymyr Mazorchuk, Rafael Mrđen. 2020-08-20. Bigrassmannian permutations and Verma modules. https://doi.org/10.1007/s00029-021-00672-z

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