A basis construction for Free subarrangements of Shi arrangements

Ruimei Gao; Meihui Jiang; Ruimei Gao; Meihui Jiang

doi:10.3934/era.2025269

Electronic Research Archive

2025, Volume 33, Issue 10: 6058-6069. doi: 10.3934/era.2025269

Previous Article Next Article

Research article

A basis construction for Free subarrangements of Shi arrangements

Ruimei Gao ,
Meihui Jiang ^,

School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130022, China

Received: 06 March 2025 Revised: 04 July 2025 Accepted: 09 October 2025 Published: 16 October 2025

D. Suyama and H. Terao established an exact basis construction for the derivation modules of the cone over the Shi arrangements utilizing Bernoulli polynomials. In this paper, building on the above basis, we introduce an explicit basis construction for a class of Free arrangements that lie between the cone of Linial arrangements and Shi arrangements.
- hyperplane arrangement,
- Shi arrangement,
- Free arrangement,
- Bernoulli polynomial,
- subarrangement
Citation: Ruimei Gao, Meihui Jiang. A basis construction for Free subarrangements of Shi arrangements[J]. Electronic Research Archive, 2025, 33(10): 6058-6069. doi: 10.3934/era.2025269

Related Papers:

Abstract

D. Suyama and H. Terao established an exact basis construction for the derivation modules of the cone over the Shi arrangements utilizing Bernoulli polynomials. In this paper, building on the above basis, we introduce an explicit basis construction for a class of Free arrangements that lie between the cone of Linial arrangements and Shi arrangements.

References

[1]	K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 265–291.
[2]	H. Terao, Multiderivations of Coxeter arrangements, Invent. Math., 148 (2002), 659–674. https://doi.org/10.1007/s002220100209 doi: 10.1007/s002220100209
[3]	J. Y. Shi, The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups, Springer-Verlag, Berlin, 2006.
[4]	T. Abe, H. Terao, Simple-root bases for Shi arrangements, J. Algebra, 422 (2015), 89–104. https://doi.org/10.1016/j.jalgebra.2014.09.011 doi: 10.1016/j.jalgebra.2014.09.011
[5]	D. Armstrong, B. Rhoades, The Shi arrangement and the Ish arrangement, Trans. Am. Math. Soc., 364 (2012), 1509–1528. https://doi.org/10.1090/s0002-9947-2011-05521-2 doi: 10.1090/s0002-9947-2011-05521-2
[6]	Z. X. Wang, G. F. Jiang, Free subarrangements of Shi arrangements, Graphs Comb., 38 (2022), 59–67. https://doi.org/10.1007/s00373-021-02399-2 doi: 10.1007/s00373-021-02399-2
[7]	M. Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math., 157 (2004), 449–454. https://doi.org/10.1007/s00222-004-0359-2 doi: 10.1007/s00222-004-0359-2
[8]	T. Abe, H. Terao, The freeness of Shi-Catalan arrangements, Eur. J. Comb., 32 (2011), 1191–1198. https://doi.org/10.1016/j.ejc.2011.06.005 doi: 10.1016/j.ejc.2011.06.005
[9]	D. Suyama, H. Terao, The Shi arrangements and the Bernoulli polynomials, Bull. London Math. Soc., 44 (2012), 563–570. https://doi.org/10.1112/blms/bdr118 doi: 10.1112/blms/bdr118
[10]	D. Suyama, A basis construction for the Shi arrangement of the type $B_\ell$ or $C_\ell$, Commun. Algebra, 43 (2015), 1435–1448. https://doi.org/10.1080/00927872.2013.865051 doi: 10.1080/00927872.2013.865051
[11]	R. M. Gao, D. H. Pei, H. Terao, The Shi arrangement of the type $D_\ell$, Proc. Japan Acad. Ser. A Math. Sci., 88 (2012), 41–45. https://doi.org/10.3792/pjaa.88.41 doi: 10.3792/pjaa.88.41
[12]	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. https://doi.org/10.1016/j.jalgebra.2017.09.024 doi: 10.1016/j.jalgebra.2017.09.024
[13]	D. Suyama, M. Yoshinaga, The primitive derivation and discrete integrals, SIGMA, 17 (2021), 563–570. https://doi.org/10.3842/SIGMA.2021.038 doi: 10.3842/SIGMA.2021.038
[14]	P. Orlik, H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)