Abstract
Let \(G\) be a simple graph on the vertex set \(\{v_1,\dots ,v_n\}\) with edge set \(E\). Let \(K\) be a field. The graphical arrangement \({\mathcal {A}}_G\) in \(K^n\) is the arrangement \(x_i\!-\!x_j\!=\!0, v_iv_j \in E\). An arrangement \({\mathcal {A}}\) is supersolvable if the intersection lattice \(L(c({\mathcal {A}}))\) of the cone \(c({\mathcal {A}})\) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement \({\mathcal {A}}_G\) is supersolvable if and only if \(G\) is a chordal graph. He later considered a generalization of graphical arrangements which are called \(\psi \)-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a \(\psi \)-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free \(\psi \)-graphical arrangements.
Similar content being viewed by others
References
Dirac, G.A.: On rigid circuit graphs. Abh. Math. Semin., Univ. Hamburg 25, 71–76 (1961)
Edelman, P., Reiner, V.: Free hyperplane arrangements between \(A_{n-1}\) and \(B_n\). Math. Z. 215, 347–365 (1994)
Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin (1992)
Stanley, R.: Supersolvable lattices. Algebra Univers. 2, 197–217 (1972)
Stanley, R.: An introduction to hyperplane arrangements. In: Miller, E., Reiner, V., Sturmfels, B. (eds.) Geometric Combinatorics. IAS/Park City Mathematics Series, vol. 13, pp. 389–396. American Mathematical Society, Providence, RI (2007)
Stanley, R.: Enumerative Combinatorics, vol. 1, second edn. Cambridge University Press, Cambridge (2012)
Stanley, R.: Valid orderings of real hyperplane arrangements. Discrete Comput. Geom. (2015). doi:10.1007/s00454-015-9683-0
Yoshinaga, M.: On the freeness of three-arrangements. Bull. Lond. Math. Soc. 37, 126–134 (2005)
Acknowledgments
The first author would like to thank the China Scholarship Council for financial support. Her work was done during her visit to the Department of Mathematics, Massachusetts Institute of Technology. The second author was partially supported by the National Science Foundation under Grant No. DMS-1068625.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Günter M. Ziegler
Rights and permissions
About this article
Cite this article
Mu, L., Stanley, R.P. Supersolvability and Freeness for \(\psi \)-Graphical Arrangements. Discrete Comput Geom 53, 965–970 (2015). https://doi.org/10.1007/s00454-015-9684-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-015-9684-z