Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-18T09:59:30.613Z Has data issue: false hasContentIssue false
  • English
  • Français

NORMALITY AND QUADRATICITY FOR SPECIAL AMPLE LINE BUNDLES ON TORIC VARIETIES ARISING FROM ROOT SYSTEMS

Published online by Cambridge University Press:  01 October 2013

QËNDRIM R. GASHI  and
TRAVIS SCHEDLER
QËNDRIM R. GASHI
Affiliation:
Department of Mathematics, University of Prishtina, Pristina 10000, Kosovo e-mail: [email protected]
TRAVIS SCHEDLER
Affiliation:
Department of Mathematics, The University of Texas at Austin 2515 Speedway, Austin, TX 78712-1202, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semi-group rings are quadratic.

Keywords

17B2214M25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue A: New developments in Noncommutative Algebra and its applications. A conference in honour of Kenny Brown's and Toby Stafford's 60th birthdays. , October 2013 , pp. 113 - 134
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Arthur, J., A local trace formula, Inst. Hautes Études Sci. Publ. Math. 73 (1991), 5–96.Google Scholar
2.Bourbaki, N., Elements of mathematics, Lie groups and Lie algebras, chapters 4–6, (Springer-Verlag, Berlin, Germany, 2002).CrossRefGoogle Scholar
3.Carrell, J. and Kurth, A., Normality of torus orbit closures in G/P, J. Algebra 233 (1) (2000), 122–134.Google Scholar
4.Carrell, J. and Kuttler, J., Smooth points of t-stable varieties in G/B and the Peterson map, Invent. Math. 151 (2) (2003), 353–379.Google Scholar
5.Dabrowski, R., On normality of the closure of a generic torus orbit in G/P, Pacific J. Math. 172 (2) (1996), 321–330.CrossRefGoogle Scholar
6.Donnelly, R. G. and Eriksson, K., The numbers game and Dynkin diagram classification results (2008) arXiv:0810.5371.Google Scholar
7.Eriksson, K., Convergence of Mozes's game of numbers, Linear Algebra Appl. 166 (1992), 151–165.Google Scholar
8.Eriksson, K., Strongly convergent games and Coxeter groups, PhD thesis, (KTH, Stockholm, Sweden, 1993).Google Scholar
9.Eriksson, K., Node firing games on graphs, Jerusalem combinatorics '93: an international conference in combinatorics (May 9–17, 1993, Jerusalem, Israel), vol. 178 (American Mathematical Society, 1994), 117–128.Google Scholar
10.Eriksson, K., Reachability is decidable in the numbers game, Theoret. Comput. Sci. 131 (1994), 431–439.Google Scholar
11.Eriksson, K., The numbers game and Coxeter groups, Discrete Math. 139 (1995), 155–166.Google Scholar
12.Eriksson, K., Strong convergence and a game of numbers, Eur. J. Combin. 17 (4) (1996), 379–390.Google Scholar
13.Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, no.131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
14.Gashi, Q. R., A vanishing result for toric varieties associated with root systems, Albanian J. Math. 4 (2007), 235–244.Google Scholar
15.Gashi, Q. R., The conjecture of Kottwitz and Rapoport in the case of split groups, PhD thesis, (The University of Chicago, Chicago, IL, June 2008).Google Scholar
16.Gashi, Q. R., The Conjecture of Kottwitz and Rapoport in the case of split groups (2008); arXiv:0805.4575.Google Scholar
17.Gashi, Q. R., Vanishing results for toric varieties associated to GL n and G 2, Transform. Groups 13 (1) (2008), 149–171.Google Scholar
18.Gashi, Q. R. and Schedler, T., On dominance and minuscule Weyl group elements J. Algebraic Combin. 33 (3) (2011), 383–399.Google Scholar
19.Benjamin, J.Howard, Matroids and geometric invariant theory of torus actions on flag spaces, J. Algebra 312 (1) (2007), 527–541.Google Scholar
20.Klyachko, A., Toric varieties and flag spaces, Teor. Chisel, Algebra i Algebr. Geom., no. 208 (Trudy Mat. Inst. Steklova, Moscow, Russia, 1995), 139–162.Google Scholar
21.Kottwitz, R. E., Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic analysis, the trace formula, and Shimura varieties (Arthur, J.et al., Editors), Clay Math. Proc., no. 4 (American Mathematical Society, Providence, RI, 2005), 393–522.Google Scholar
22.Mozes, S., Reflection processes on graphs and Weyl groups, J. Combin. Theory Ser. A 53 (1) (1990), 128–142.Google Scholar
23.Payne, S., Lattice polytopes cut out by root systems and the Koszul property, Adv. Math. 220 (3) (2009), 926–935.Google Scholar
24.Procesi, C., The toric variety associated to Weyl chambers (English summary) Mots (Lang. Raison. Calc., Hermès, Paris, France, 1990), 153–161.Google Scholar
25.Proctor, R. A., Bruhat lattices, plane partition generating functions, and minuscule representations, Euro. J. Combin. 5 (1984), 331–350.Google Scholar
26.Proctor, R. A., Minuscule elements of Weyl groups, the numbers game, and d-complete posets, J. Algebra 213 (1999), 272–303.Google Scholar
27.Rapoport, M., A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2) (2000), 153–166.Google Scholar
28.Stembridge, J. R., The partial order of dominant weights, Adv. Math. 136 (1998), 340–364.CrossRefGoogle Scholar
29.Sturmfels, B., Gröbner bases and convex polytopes, University Lecture Series, vol. 8, (American Mathematical Society, Providence, RI, 1996).Google Scholar
30.Wildberger, N. J., A combinatorial construction for simply-laced Lie algebras, Adv. in Appl. Math. 30 (2003), 385–396.Google Scholar
31.Wildberger, N. J., Minuscule posets from neighbourly graph sequences, Euro. J. Combin. 24 (2003), 741–757.Google Scholar