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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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  • [1] T. Abe, M. Barakat, M. Cuntz, T. Hoge, H. Terao, The freeness of ideal subarrangements of Weyl arrangements, J. Eur. Math. Soc., to appear.
  • [2] Hélène Barcelo and Edwin Ihrig, Lattices of parabolic subgroups in connection with hyperplane arrangements, J. Algebraic Combin. 9 (1999), no. 1, 5-24. MR 1676736 (2000g:52023), https://doi.org/10.1023/A:1018607830027
  • [3] Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010), https://doi.org/10.1090/S0894-0347-96-00192-0
  • [4] Anders Björner, Paul H. Edelman, and Günter M. Ziegler, Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom. 5 (1990), no. 3, 263-288. MR 1036875 (90k:51036), https://doi.org/10.1007/BF02187790
  • [5] Graham Denham and Sergey Yuzvinsky, Annihilators of Orlik-Solomon relations, Adv. in Appl. Math. 28 (2002), no. 2, 231-249. MR 1888846 (2003b:05046), https://doi.org/10.1006/aama.2001.0779
  • [6] Michael Falk, Line-closed matroids, quadratic algebras, and formal arrangements, Adv. in Appl. Math. 28 (2002), no. 2, 250-271. MR 1888847 (2003a:05040), https://doi.org/10.1006/aama.2001.0780
  • [7] R. Fröberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997) Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, New York, 1999, pp. 337-350. MR 1767430 (2001i:16046)
  • [8] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [9] Michel Jambu and Stefan Papadima, A generalization of fiber-type arrangements and a new deformation method, Topology 37 (1998), no. 6, 1135-1164. MR 1632975 (99g:52019), https://doi.org/10.1016/S0040-9383(97)00092-X
  • [10] Dinh Van Le and Tim Römer, Broken circuit complexes and hyperplane arrangements, J. Algebraic Combin. 38 (2013), no. 4, 989-1016. MR 3119368, https://doi.org/10.1007/s10801-013-0435-z
  • [11] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167-189. MR 558866 (81e:32015), https://doi.org/10.1007/BF01392549
  • [12] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
  • [13] Henry K. Schenck and Alexander I. Suciu, Lower central series and free resolutions of hyperplane arrangements, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3409-3433 (electronic). MR 1911506 (2003k:52022), https://doi.org/10.1090/S0002-9947-02-03021-0
  • [14] Brad Shelton and Sergey Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. (2) 56 (1997), no. 3, 477-490. MR 1610447 (99c:16044), https://doi.org/10.1112/S0024610797005553
  • [15] Eric Sommers and Julianna Tymoczko, Exponents for $ B$-stable ideals, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3493-3509. MR 2218986 (2007a:17016), https://doi.org/10.1090/S0002-9947-06-04080-3
  • [16] R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197-217. MR 0309815 (46 #8920)
  • [17] Hiroaki Terao, Arrangements of hyperplanes and their freeness. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 293-312. MR 586451 (84i:32016a)
  • [18] S. Yuzvinskiĭ, Orlik-Solomon algebras in algebra and topology, Uspekhi Mat. Nauk 56 (2001), no. 2(338), 87-166 (Russian, with Russian summary); English transl., Russian Math. Surveys 56 (2001), no. 2, 293-364. MR 1859708 (2002i:14047), https://doi.org/10.1070/RM2001v056n02ABEH000383

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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

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