Supersolvability and the Koszul property of root ideal arrangements

Hultman, Axel

doi:10.1090/proc/12810

Supersolvability and the Koszul property of root ideal arrangements
HTML articles powered by AMS MathViewer

by Axel Hultman PDF

Proc. Amer. Math. Soc. 144 (2016), 1401-1413 Request permission

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

T. Abe, M. Barakat, M. Cuntz, T. Hoge, H. Terao, The freeness of ideal subarrangements of Weyl arrangements, J. Eur. Math. Soc., to appear.
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, DOI 10.1023/A:1018607830027
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, DOI 10.1090/S0894-0347-96-00192-0
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, DOI 10.1007/BF02187790
Graham Denham and Sergey Yuzvinsky, Annihilators of Orlik-Solomon relations, Adv. in Appl. Math. 28 (2002), no. 2, 231–249. MR 1888846, DOI 10.1006/aama.2001.0779
Michael Falk, Line-closed matroids, quadratic algebras, and formal arrangements, Adv. in Appl. Math. 28 (2002), no. 2, 250–271. MR 1888847, DOI 10.1006/aama.2001.0780
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
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
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, DOI 10.1016/S0040-9383(97)00092-X
Dinh Van Le and Tim Römer, Broken circuit complexes and hyperplane arrangements, J. Algebraic Combin. 38 (2013), no. 4, 989–1016. MR 3119368, DOI 10.1007/s10801-013-0435-z
Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189. MR 558866, DOI 10.1007/BF01392549
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, DOI 10.1007/978-3-662-02772-1
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. MR 1911506, DOI 10.1090/S0002-9947-02-03021-0
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, DOI 10.1112/S0024610797005553
Eric Sommers and Julianna Tymoczko, Exponents for $B$-stable ideals, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3493–3509. MR 2218986, DOI 10.1090/S0002-9947-06-04080-3
R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197–217. MR 309815, DOI 10.1007/BF02945028
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
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, DOI 10.1070/RM2001v056n02ABEH000383