Mathematical Research Letters

Volume 26 (2019)

Number 2

Indecomposable manipulations with simple modules in category $\mathcal{O}$

Pages: 447 – 499

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n2.a5

Authors

Kevin Coulembier (School of Mathematics and Statistics, University of Sydney, Australia)

Volodymyr Mazorchuk (Department of Mathematics, University of Uppsala, Sweden)

Xiaoting Zhang (Department of Mathematics, University of Uppsala, Sweden)

Abstract

We study the problem of indecomposability of translations of simple modules in the principal block of BGG category $\mathcal{O}$ for $\mathfrak{sl}_n$, as conjectured in “Parabolic projective functors in type A” [T. Kildetoft and V. Mazorchuk, Adv. Math. 301 (2016), 785–803]. We describe some general techniques and prove a few general results which may be applied to study various special cases of this problem. We apply our results to verify indecomposability for $n \leq 6$. We also study the problem of indecomposability of shufflings and twistings of simple modules and obtain some partial results.

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The first author is supported by the Australian Research Council. The second and the third authors are supported by the Swedish Research Council and the Göran Gustafsson Foundation.

Received 14 September 2017

Accepted 1 May 2018

Published 12 August 2019