Supersolvability and Freeness for ψ-Graphical Arrangements
Let G be a simple graph on the vertex set {v[subscript 1],…,v[subscript n]} with edge set E. Let K be a field. The graphical arrangement A[subscript G] in K[superscript n] is the arrangement x[subscript i]−x[subscript j]=0,v[subscript i]v[subscript j] ∈ E. An arrangement A is supersolvable...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer US
2016
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| Online Access: | http://hdl.handle.net/1721.1/104653 https://orcid.org/0000-0003-3123-8241 |
| _version_ | 1826208674708717568 |
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| author | Mu, Lili Stanley, Richard P |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Mu, Lili Stanley, Richard P |
| author_sort | Mu, Lili |
| collection | MIT |
| description | Let G be a simple graph on the vertex set {v[subscript 1],…,v[subscript n]} with edge set E. Let K be a field. The graphical arrangement A[subscript G] in K[superscript n] is the arrangement x[subscript i]−x[subscript j]=0,v[subscript i]v[subscript j] ∈ E. An arrangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement A[subscript G] is supersolvable if and only if G is a chordal graph. He later considered a generalization of graphical arrangements which are called ψ-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a ψ-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free ψ-graphical arrangements. |
| first_indexed | 2024-09-23T14:09:52Z |
| format | Article |
| id | mit-1721.1/104653 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:09:52Z |
| publishDate | 2016 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1046532022-09-28T18:59:16Z Supersolvability and Freeness for ψ-Graphical Arrangements Mu, Lili Stanley, Richard P Massachusetts Institute of Technology. Department of Mathematics Mu, Lili Stanley, Richard P Let G be a simple graph on the vertex set {v[subscript 1],…,v[subscript n]} with edge set E. Let K be a field. The graphical arrangement A[subscript G] in K[superscript n] is the arrangement x[subscript i]−x[subscript j]=0,v[subscript i]v[subscript j] ∈ E. An arrangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement A[subscript G] is supersolvable if and only if G is a chordal graph. He later considered a generalization of graphical arrangements which are called ψ-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a ψ-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free ψ-graphical arrangements. China Scholarship Council National Science Foundation (U.S.) (Grant DMS-1068625) 2016-10-06T19:49:09Z 2016-10-06T19:49:09Z 2015-04 2015-01 2016-08-18T15:41:14Z Article http://purl.org/eprint/type/JournalArticle 0179-5376 1432-0444 http://hdl.handle.net/1721.1/104653 Mu, Lili, and Richard P. Stanley. “Supersolvability and Freeness for ψ-Graphical Arrangements.” Discrete & Computational Geometry 53.4 (2015): 965–970. https://orcid.org/0000-0003-3123-8241 en http://dx.doi.org/10.1007/s00454-015-9684-z Discrete & Computational Geometry Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Mu, Lili Stanley, Richard P Supersolvability and Freeness for ψ-Graphical Arrangements |
| title | Supersolvability and Freeness for ψ-Graphical Arrangements |
| title_full | Supersolvability and Freeness for ψ-Graphical Arrangements |
| title_fullStr | Supersolvability and Freeness for ψ-Graphical Arrangements |
| title_full_unstemmed | Supersolvability and Freeness for ψ-Graphical Arrangements |
| title_short | Supersolvability and Freeness for ψ-Graphical Arrangements |
| title_sort | supersolvability and freeness for ψ graphical arrangements |
| url | http://hdl.handle.net/1721.1/104653 https://orcid.org/0000-0003-3123-8241 |
| work_keys_str_mv | AT mulili supersolvabilityandfreenessforpsgraphicalarrangements AT stanleyrichardp supersolvabilityandfreenessforpsgraphicalarrangements |