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Supersolvability and the Koszul property of root ideal arrangements


Author: Axel Hultman
Journal: Proc. Amer. Math. Soc. 144 (2016), 1401-1413
MSC (2010): Primary 05B35; Secondary 20F55, 16S37
DOI: https://doi.org/10.1090/proc/12810
Published electronically: September 9, 2015
MathSciNet review: 3451219
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Abstract: A root ideal arrangement $ \mathcal {A}_I$ is the set of reflecting hyperplanes corresponding to the roots in an order ideal $ I\subseteq \Phi ^+$ of the root poset on the positive roots of a finite crystallographic root system $ \Phi $. A characterisation of supersolvable root ideal arrangements is obtained. Namely, $ \mathcal {A}_I$ is supersolvable if and only if $ I$ is chain peelable, meaning that it is possible to reach the empty poset from $ I$ by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved undertaking subideals. We identify the minimal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type $ D_4$ and one in type $ F_4$. By showing that $ \mathcal {A}_I$ is not line-closed if $ I$ contains one of these, we deduce that the Orlik-Solomon algebra $ \mathcal {OS}({\mathcal {A}_I})$ has the Koszul property if and only if $ \mathcal {A}_I$ is supersolvable.


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Additional Information

Axel Hultman
Affiliation: Department of Mathematics, Linköping University, SE-581 83, Linköping, Sweden
Email: axel.hultman@liu.se

DOI: https://doi.org/10.1090/proc/12810
Received by editor(s): October 14, 2014
Received by editor(s) in revised form: April 8, 2015
Published electronically: September 9, 2015
Communicated by: Patricia Hersh
Article copyright: © Copyright 2015 American Mathematical Society