Supersolvability and the Koszul property of root ideal arrangements
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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.References
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Additional Information
- Axel Hultman
- Affiliation: Department of Mathematics, Linköping University, SE-581 83, Linköping, Sweden
- Email: axel.hultman@liu.se
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1401-1413
- MSC (2010): Primary 05B35; Secondary 20F55, 16S37
- DOI: https://doi.org/10.1090/proc/12810
- MathSciNet review: 3451219