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Young diagrams and intersection numbers for toric manifolds associated with Weyl chambers. (English) Zbl 1333.14045

Let \(\Phi \) be a root system in the \(n-\)dimensional Euclidean space \(E\) with its inner product. Let \(\Delta (\Phi )\) be the fan determined by the collection of the Weyl chambers in \(E^*\), and \(X\) the toric manifold associated to \(\Delta (\Phi )\). The Weyl group naturally acts on \(X\).
The aim of this paper is to study the intersection numbers of invariant divisors in \(X\). One of the main results is a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. It turns out that these numbers describe the ring structure of the cohomology of \(X\).

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
17B22 Root systems
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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References:

[1] V. Batyrev and Mark Blume, {\it The functor for toric varieties associated with Weyl} {\it chambers and Losev-Manin moduli spaces}, Tohoku Math. J. 63 (2011), 581-604. · Zbl 1255.14041
[2] A. Bj¨orner and F. Brenti, {\it Combinatorics of Coxeter groups}, Graduate Texts in Mathematics 231, Springer, New York, 2005. · Zbl 1110.05001
[3] D. Cox, J. Little and H. Schenck, {\it Toric varieties}, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence RI (2011). · Zbl 1223.14001
[4] F. De Mari, C. Procesi and M. A. Shayman, {\it Hessenberg varieties}, Trans. Amer. Math. Soc. 332 (1992), no. 2, 529-534. · Zbl 0770.14022
[5] V. Dolgachev and V. Lunts, {\it A character formula for the representation of a Weyl} {\it group in the cohomology of the associated toric variety}, J. Algebra 168 (1994), 741772. · Zbl 0813.14040
[6] W. Fulton, {\it An Introduction to Toric Varieties}, Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton NJ, 1993. · Zbl 0813.14039
[7] A. Klyachko, {\it Orbits of a maximal torus on a flag space}, Functional Anal. Appl. 19 (1985), no. 2, 65-66. · Zbl 0581.14038
[8] A. Losev and Y. Manin, {\it New moduli spaces of pointed curves and pencils of flat} {\it connections}, Michigan Math. J. 48 (2000), 443-472. · Zbl 1078.14536
[9] C. Procesi, {\it The toric variety associated to Weyl chambers}, Mots, 153-161, Lang. Raison. Calc., Herm‘es, Paris, 1990. · Zbl 1177.14090
[10] J. Stembridge, {\it Some permutation representations of Weyl groups associated with the} {\it cohomology of toric varieties}, Adv. Math. 106 (1994), 244-301. · Zbl 0838.20050
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