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Transactions of the American Mathematical Society

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Computation of Kazhdan-Lusztig polynomials and some applications to finite groups


Author: Frank Lübeck
Journal: Trans. Amer. Math. Soc. 373 (2020), 2331-2347
MSC (2010): Primary 20F55, 20C08, 20G05, 20G10, 20E28
DOI: https://doi.org/10.1090/tran/8037
Published electronically: January 28, 2020
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Abstract: We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig.

Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases.

Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conjecture by G. E. Wall.


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Additional Information

Frank Lübeck
Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen, Pontdriesch 14/16, D-52064 Aachen, Germany
Email: frank.luebeck@math.rwth-aachen.de

DOI: https://doi.org/10.1090/tran/8037
Received by editor(s): October 22, 2016
Received by editor(s) in revised form: August 4, 2017, and March 25, 2019
Published electronically: January 28, 2020
Additional Notes: The author acknowledges support from the DFG within the TRR 195
Article copyright: © Copyright 2020 American Mathematical Society