The purpose of this paper is twofold. Firstly, we generalize the notion of semigeneric braid arrangement to semigeneric graphical arrangement. Secondly, we give a formula for the characteristic polynomial of the semigeneric graphical arrangement $ \mathcal{S}_G: = \{x_i- x_j = a_i: ij\in E(G), 1\leq i < j\leq n \} $, where $ a_i{\rm{'s}} $ are generic. The formula is obtained by characterizing central subarrangements of $ \mathcal{S}_{G} $ via the corresponding graph $ {G} $.
Citation: Ruimei Gao, Jingjing Qiang, Miao Zhang. The Characteristic polynomials of semigeneric graphical arrangements[J]. AIMS Mathematics, 2023, 8(2): 3226-3235. doi: 10.3934/math.2023166
The purpose of this paper is twofold. Firstly, we generalize the notion of semigeneric braid arrangement to semigeneric graphical arrangement. Secondly, we give a formula for the characteristic polynomial of the semigeneric graphical arrangement $ \mathcal{S}_G: = \{x_i- x_j = a_i: ij\in E(G), 1\leq i < j\leq n \} $, where $ a_i{\rm{'s}} $ are generic. The formula is obtained by characterizing central subarrangements of $ \mathcal{S}_{G} $ via the corresponding graph $ {G} $.
| [1] |
T. Abe, Roots of characteristic polynomials and intersection points of line arrangements, J. Singularities, 8 (2014), 100–117. http://dx.doi.org/10.5427/JSING.2014.8H doi: 10.5427/JSING.2014.8H
|
| [2] |
T. Abe, Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations, Topology Appl., 313 (2022), 107990. http://dx.doi.org/10.1016/j.topol.2021.107990 doi: 10.1016/j.topol.2021.107990
|
| [3] |
T. Abe, D. Suyama, A basis construction of the extended Catalan and Shi arrangements of the type $A_2$, J. Algebra, 493 (2018), 20–35. http://dx.doi.org/10.1016/J.JALGEBRA.2017.09.024 doi: 10.1016/J.JALGEBRA.2017.09.024
|
| [4] |
T. Abe, H. Terao, Simple-root bases for Shi arrangements, J. Algebra, 422 (2015), 89–104. http://dx.doi.org/10.1016/j.jalgebra.2014.09.011 doi: 10.1016/j.jalgebra.2014.09.011
|
| [5] |
T. Abe, H. Terao, The freeness of Shi-Catalan arrangements, European J. Combin., 32 (2011), 1191–1198. http://dx.doi.org/10.1016/j.ejc.2011.06.005 doi: 10.1016/j.ejc.2011.06.005
|
| [6] |
T. Abe, M. Yoshinaga, Free arrangements and coefficients of characteristic polynomials, Math. Z., 275 (2013), 911–919. http://dx.doi.org/10.1007/S00209-013-1165-6 doi: 10.1007/S00209-013-1165-6
|
| [7] |
C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math., 122 (1996), 193–233. http://dx.doi.org/10.1006/AIMA.1996.0059 doi: 10.1006/AIMA.1996.0059
|
| [8] |
O. Bernardi, Deformations of the braid arrangement and trees, Adv. Math., 335 (2018), 466–518. http://dx.doi.org/10.1016/J.AIM.2018.07.020 doi: 10.1016/J.AIM.2018.07.020
|
| [9] |
M. D'Adderio, L. Moci, Arithmetic matroids, the Tutte polynomial and toric arrangements, Adv. Math., 232 (2013), 335–367. http://dx.doi.org/10.1016/J.AIM.2012.09.001 doi: 10.1016/J.AIM.2012.09.001
|
| [10] |
C. D. Concini, C. Procesi, On the geometry of toric arrangements, Transform. Groups, 10 (2005), 387–422. http://dx.doi.org/10.1007/S00031-005-0403-3 doi: 10.1007/S00031-005-0403-3
|
| [11] |
R. M. Gao, D. H. Pei, H. Terao, The Shi arrangement of the type $D_\ell$, P. Jan. Acad. A-Math., 88 (2012), 41–45. http://dx.doi.org/10.3792/pjaa.88.41 doi: 10.3792/pjaa.88.41
|
| [12] |
H. Kamiya, A. Takemura, H. Terao, Periodicity of hyperplane arrangements with integral coefficients modulo positive integers, J. Algebraic Combin., 27 (2008), 317–330. http://dx.doi.org/10.1007/S10801-007-0091-2 doi: 10.1007/S10801-007-0091-2
|
| [13] |
J. Lawrence, Enumeration in torus arrangements, European J. Combin., 32 (2011), 870–881. http://dx.doi.org/10.1016/j.ejc.2011.02.003 doi: 10.1016/j.ejc.2011.02.003
|
| [14] |
Y. Liu, T. N. Tran, M. Yoshinaga, G-Tutte polynomials and abelian Lie group arrangements, Int. Math. Res. Notices, 2021 (2021), 150–188. http://dx.doi.org/10.1093/imrn/rnz092 doi: 10.1093/imrn/rnz092
|
| [15] |
L. Moci, A Tutte polynomial for toric arrangements, T. Am. Math. Soc., 364 (2012), 1067–1088. http://dx.doi.org/10.1090/S0002-9947-2011-05491-7 doi: 10.1090/S0002-9947-2011-05491-7
|
| [16] |
P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplane, Invent. Math., 56 (1980), 167–189. http://dx.doi.org/10.1007/BF01392549 doi: 10.1007/BF01392549
|
| [17] | P. Orlik, H. Terao, Arrangements of hyperplanes, Berlin: Springer-Verlag, 1992. http://dx.doi.org/10.1007/978-3-662-02772-1 |
| [18] |
R. P. Stanley, An introduction to hyperplane arrangements, Geom. Combin., 13 (2007), 389–496. http://dx.doi.org/10.1090/pcms/013/08 doi: 10.1090/pcms/013/08
|
| [19] |
D. Suyama, H. Terao, The Shi arrangements and the Bernoulli polynomials, B. Lond. Math. Soc., 44 (2012), 563–570. http://dx.doi.org/10.1112/blms/bdr118 doi: 10.1112/blms/bdr118
|
| [20] |
H. Terao, Multiderivations of Coxeter arrangements, Invent. Math., 148 (2002), 659–674. http://dx.doi.org/10.1007/s002220100209 doi: 10.1007/s002220100209
|
| [21] |
T. N. Tran, M. Yoshinaga, Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials, J. Comb. Theory A, 165 (2019), 258–272. http://dx.doi.org/10.1016/j.jcta.2019.02.003 doi: 10.1016/j.jcta.2019.02.003
|
| [22] |
W. T. Tutte, A contribution to the theory of chromatic polynomials, Can. J. Math., 6 (1954), 80–91. http://dx.doi.org/10.4153/CJM-1954-010-9 doi: 10.4153/CJM-1954-010-9
|
| [23] |
M. Yoshinaga, Characteristic polynomials of Linial arrangements for exceptional root systems, J. Comb. Theory A, 157 (2018), 267–286. http://dx.doi.org/10.1016/j.jcta.2018.02.011 doi: 10.1016/j.jcta.2018.02.011
|
| [24] |
M. Yoshinaga, Worpitzky partitions for root systems and characteristic quasi-polynomials, Tohoku Math. J., 70 (2018), 39–63. http://dx.doi.org/10.2748/TMJ/1520564418 doi: 10.2748/TMJ/1520564418
|
Figures(3) / Tables(2)
Ruimei Gao, Jingjing Qiang, Miao Zhang. The Characteristic polynomials of semigeneric graphical arrangements[J]. AIMS Mathematics, 2023, 8(2): 3226-3235. doi: 10.3934/math.2023166
DownLoad: