JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2014, Vol. 49 ›› Issue (12): 66-70.doi: 10.6040/j.issn.1671-9352.0.2014.187

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The freeness of Shi-Catalan arrangements of type G2

GAO Rui-mei   

  1. Department of Science, Changchun University of Science and Technology, Changchun 130022, Jilin, China
  • Received:2014-04-28 Revised:2014-09-09 Online:2014-12-20 Published:2014-12-20

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Abstract: The Shi-Catalan arrangements of type G2 are multiarrangements in the 2-dimensional vector space, which are obtained by giving the same multiplicity to the hyperplanes in the same orbit in Weyl arrangements of type G2. Four concrete forms of Shi-Catalan arrangements of type G2 are given. By counting the intersections of hyperplanes in reflection planes, we prove the conclusion that the cones over the Shi-Catalan arrangements of type G2 are all free.

Key words: Weyl arrangement, freeness, Shi-Catalan arrangement

CLC Number: 

  • O189
[1] ORLIK P, TERAO H. Arrangements of hyperplanes[M]// Grundlehren der Mathematischen Wissenschaften, 300. Berlin: Springer-Verlag, 1992: 1-325.
[2] ORLIK P, SOLOMON L. Combinatorics and topology of complements of hyperplanes[J]. Inventiones Mathematicae, 1980, 56:167-189.
[3] JIANG Guangfeng, YU Jianming. Supersolvability of complementary signed-graphic hyperplane arrangements[J]. The Australasian Journal of Combinatorics, 2004, 30:261-274.
[4] ABE T, TERAO H, WAKEFIELD M. The characteristic polynomial of a multiarrangement[J]. Advances in Mathematics, 2007, 215: 825-838.
[5] YUZVINSKY S. Orlik-Solomon algebras in algebra and topology[J]. Russian Mathematical Surveys, 2001, 56: 293-364.
[6] GAO Ruimei, PEI Donghe. The Supersolvable order of hyperplanes of an arrangement[J]. Communications in Mathematical Research, 2013, 29(3):231-238.
[7] 高瑞梅, 裴东河. 构形的特征多项式和超可解性的算法[J]. 山东大学学报: 理学版, 2014, 49(2):51-57. GAO Ruimei, PEI Donghe. The algorithms of characteristic polynomial and supersolvability of a hyperplane arrangement[J]. Journal of Shandong University: Natural Science, 2014, 49(2):51-57.
[8] YOSHINAGA M. Characterization of a free arrangement and conjecture of Edelman and Reiner[J]. Inventiones Mathematicae, 2004, 157(2):449-454.
[9] SUYAMA D, TERAO H. The Shi arrangements and the Bernoulli polynomials[J]. The Bulletin of the London Mathematical Society, 2012, 44:563-570.
[10] GAO Ruimei, PEI Donghe, TERAO H. The Shi arrangement of the type Dl[KG-*4][J]. Proceedings of the Japan Academy, Series A: Mathematical Sciences, 2012, 88:41-45.
[11] SHI Jianyi. The Kazhdan-Lusztig cells in certain affine Weyl groups[M]// Lecture Notes in Mathematics, 1179. New York: Springer-Verlag, 1986.
[12] ABE T, TERAO H. The freeness of Shi-Catalan arrangements[J]. European Journal of Combinatorics, 2011, 32:1191-1198.
[1] GAO Rui-mei, CHU Ying. Freeness of arrangements between the Weyl arrangements of types An-1 and Bn [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 70-75.
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