Cohomology of complements of toric arrangements associated with root systems. (English) Zbl 1490.14087
Summary: We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group \(\Gamma\) is acting on the arrangement, the algorithm determines the cohomology groups as representations of \(\Gamma\). As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14N20 | Configurations and arrangements of linear subspaces |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Software:
SageMathReferences:
| [1] | Ardila, F., Castillo, F., Henley, M.: The arithmetic Tutte polynomials of the classical root systems. Int. Math. Res. Not. IMRN 12, 3830-3877 (2015) · Zbl 1317.05091 |
| [2] | Arnol’d, V., The cohomology ring of the colored braid group, Mat. Zametki, 5, 227-231 (1969) · Zbl 0277.55002 |
| [3] | Bergström, J., Bergvall, O.: The equivariant Euler characteristic of \(\cal{A}_3[2]\). Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XX, 1345-1357 (2020) · Zbl 1483.14081 |
| [4] | Bergvall, O.: Cohomology of arrangements and moduli spaces. Ph.D. thesis, Stockholms Universitet (2016) |
| [5] | Bergvall, O., Cohomology of the moduli space of genus three curves with symplectic level two structure, Geom. Dedic., 202, 1, 165-191 (2019) · Zbl 1423.14182 · doi:10.1007/s10711-018-0407-5 |
| [6] | Bergvall, O., Cohomology of the toric arrangementassociated with \(A_n\), J. Fixed Point Theory Appl., 21, 15 (2019) · Zbl 1454.20081 · doi:10.1007/s11784-018-0655-x |
| [7] | Bergvall, O., Gounelas, F.: Cohomology of moduli spaces of Del Pezzo surfaces. Math. Nachr (to appear) · Zbl 1535.14084 |
| [8] | Brändén, P.; Moci, L., The multivariate arithmetic Tutte polynomial, Trans. Am. Math. Soc., 366, 10, 5523-5540 (2014) · Zbl 1300.05133 · doi:10.1090/S0002-9947-2014-06092-3 |
| [9] | Brieskorn, E.: Sur les groupes de tresses. Séminaire Bourbaki 14 (1971-1972), pp. 21-44 · Zbl 0277.55003 |
| [10] | Carter, R.: Finite groups of Lie type, Wiley Classics Library, Wiley, Chichester (Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication) (1993) |
| [11] | Cox, D.; Little, J.; Schenck, H., Toric Varieties, Graduate Studies in Mathematics (2011), Providence: American Mathematical Society, Providence · Zbl 1223.14001 |
| [12] | Colombo, E., van Geemen, B., Looijenga, E.: Del Pezzo moduli via root systems, Algebra, arithmetic, and geometry. In: honor of Yu. I. Manin, Vol. 1 (Y. Tschinkel and Y. Zarhin, eds.), Progr. Math, vol. 269. Birkhäuser, Boston, Inc., pp. 291-337 (2009) · Zbl 1195.14051 |
| [13] | Karp, R.; Mossel, E.; Riesenfeld, S.; Verbin, E.; Daskalakis, C., Sorting and selection in posets, SIAM J. Comput., 40, 3, 392-401 (2007) · Zbl 1421.68034 |
| [14] | De Concini, C.; Procesi, C., On the geometry of toric arrangements, Transform. Groups, 10, 3-4, 387-422 (2005) · Zbl 1099.14043 · doi:10.1007/s00031-005-0403-3 |
| [15] | The Sage Developers: SageMath, the Sage Mathematics Software System (Version 6.8) (2015). http://www.sagemath.org |
| [16] | Dimca, A., Lehrer, G.: Purity and Equivariant Weight Polynomials. In: Lehrer, G. (ed.) Algebraic Groups and Lie Groups. Australian Mathematical Society Lecture Series. Cambridge University Press, pp. 161-181 (1997) · Zbl 0905.57022 |
| [17] | Eisenbud, D.; Sturmfels, B., Binomial ideals, Duke Math. J., 84, 1, 1-45 (1996) · Zbl 0873.13021 · doi:10.1215/S0012-7094-96-08401-X |
| [18] | Fleischmann, P.; Janiszczak, I., Combinatorics and Poincaré polynomials of hyperplane complements for exceptional Weyl groups, J. Combin. Theory. Ser. A, 63, 2, 257-274 (1993) · Zbl 0838.20045 · doi:10.1016/0097-3165(93)90060-L |
| [19] | Fink, A., Moci, L.: Matroids Over a Ring, Combinatorial Methods in Topology and Algebra. Springer INdAM Ser., vol. 12. Springer, Cham, pp. 41-47 (2015) · Zbl 1332.05028 |
| [20] | Gross, M.; Hacking, P.; Keel, S., Birational geometry of cluster algebras, Algebr. Geom., 2, 2, 137-175 (2015) · Zbl 1322.14032 · doi:10.14231/AG-2015-007 |
| [21] | Getzler, E., Looijenga, E.: The hodge polynomial of \(\cal{M}_{3,1} \). arxiv:math/9910174 (1999) |
| [22] | Lehrer, G.: On the Poincaré series associated with Coxeter group actions on complements of hyperplanes. J. Lond. Math. Soc. (2) 36(2), 275-294 (1987) · Zbl 0649.20041 |
| [23] | Lehrer, G., A toral configuration space and regular semisimple conjugacy classes, Math. Proc. Camb. Philos. Soc., 118, 1, 105-113 (1995) · Zbl 0839.20060 · doi:10.1017/S0305004100073497 |
| [24] | Looijenga, E.: Cohomology of \(\cal{M}_3\) and \(\cal{M}_3^1\). In: Bödigheimer, C.-F., Hain, R.M. (eds.) Mapping Class Groups and Moduli Spaces of Riemann Surfaces. Contemporary Mathematics, vol. 150, pp. 205-228 (1993) · Zbl 0814.14029 |
| [25] | Liu, Y.; Tran, TN; Yoshinaga, M., \(G\)-tutte polynomials and abelian lie group arrangements, Int. Math. Res. Not. IMRN, 1, 150-188 (2021) · Zbl 1470.05084 |
| [26] | Macmeikan, C., The Poincaré polynomial of an MP arrangement, Proc. Amer. Math. Soc., 132, 6, 1575-1580 (2004) · Zbl 1079.14026 · doi:10.1090/S0002-9939-04-07398-8 |
| [27] | Moci, L.: Combinatorics and topology of toric arrangements defined by root systems. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(4), 293-308 (2008) · Zbl 1194.14077 |
| [28] | Madsen, I., Tornehave, J.: From Calculus to Cohomology. Cambridge University Press, Cambridge (1997) · Zbl 0884.57001 |
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