Unimodular Triangulations and Coverings of Configurations Arising from Root Systems
Hidefumi Ohsugi
and Takayuki Hibi
DOI: 10.1023/A:1012772002661
Abstract
Existence of a regular unimodular triangulation of the configuration F + È{ (0,0,...,0)} \text in R n Φ^ + \cup \{ (0,0,...,0)\} {\text{ in }}R^n , where F Ì R n Φ\subset R^n and where (0, 0,...,0 ) is the origin of R n R^n , will be shown for 
= B n , C n , D n and BC n . Moreover, existence of a unimodular covering of a certain subconfiguration of the configuration A n+1 + will be studied.
= B n , C n , D n and BC n . Moreover, existence of a unimodular covering of a certain subconfiguration of the configuration A n+1 + will be studied.Pages: 199–219
Keywords: initial ideals; unimodular triangulations; unimodular coverings; root systems; positive roots
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References
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7. H. Ohsugi and T. Hibi, “Normal polytopes arising from finite graphs,” J. Algebra 207 (1998), 409-426.
8. H. Ohsugi and T. Hibi, “Koszul bipartite graphs,” Advances in Applied Math. 22 (1999), 25-28.
9. H. Ohsugi and T. Hibi, “Toric ideals generated by quadratic binomials,” J. Algebra 218 (1999), 509-527.
10. H. Ohsugi and T. Hibi, “Compressed polytopes, initial ideals and complete multipartite graphs,” Illinois J. Math. 44 (2000), 391-406.
11. H. Ohsugi and T. Hibi, “Combinatorics on the configuration of positive roots of a root system,” in preparation.
12. I. Peeva, V. Reiner, and B. Sturmfels, “How to shell a monoid,” Math. Ann. 310 (1998), 379-393.
13. R.P. Stanley, “Decompositions of rational convex polytopes,” Ann. Discrete Math. 6 (1980), 333-342.
14. R.P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge, New York, Sydney, 1999.
15. A. Simis, W.V. Vasconcelos, and R.H. Villarreal, “The integral closure of subrings associated to graphs,” J. Algebra 199 (1998), 281-289.
16. B. Sturmfels, Gr\ddot obner Bases and Convex Polytopes, Amer. Math. Soc., Providence, RI, 1995.
17. W.V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Springer- Verlag, Berlin, Heidelberg, New York, 1998.
2. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties and Algorithms, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
3. W. Fong, “Triangulations and combinatorial properties of convex polytopes,” Dessertation, M.I.T., June 2000.
4. I.M. Gelfand, M.I. Graev, and A. Postnikov, “Combinatorics of hypergeometric functions associated with positive roots,” in Arnold-Gelfand Mathematics Seminars, Geometry and Singularity Theory. V.I. Arnold, I.M. Gelfand, M. Smirnov, and V.S. Retakh (Eds.), Birkh\ddot auser, Boston, 1997, pp. 205-221.
5. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, 2nd printing, revised, Springer- Verlag, Berlin, Heidelberg, New York, 1972.
6. H. Ohsugi, J. Herzog, and T. Hibi, “Combinatorial pure subrings,” Osaka J. Math. 37 (2000), 745-757.
7. H. Ohsugi and T. Hibi, “Normal polytopes arising from finite graphs,” J. Algebra 207 (1998), 409-426.
8. H. Ohsugi and T. Hibi, “Koszul bipartite graphs,” Advances in Applied Math. 22 (1999), 25-28.
9. H. Ohsugi and T. Hibi, “Toric ideals generated by quadratic binomials,” J. Algebra 218 (1999), 509-527.
10. H. Ohsugi and T. Hibi, “Compressed polytopes, initial ideals and complete multipartite graphs,” Illinois J. Math. 44 (2000), 391-406.
11. H. Ohsugi and T. Hibi, “Combinatorics on the configuration of positive roots of a root system,” in preparation.
12. I. Peeva, V. Reiner, and B. Sturmfels, “How to shell a monoid,” Math. Ann. 310 (1998), 379-393.
13. R.P. Stanley, “Decompositions of rational convex polytopes,” Ann. Discrete Math. 6 (1980), 333-342.
14. R.P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge, New York, Sydney, 1999.
15. A. Simis, W.V. Vasconcelos, and R.H. Villarreal, “The integral closure of subrings associated to graphs,” J. Algebra 199 (1998), 281-289.
16. B. Sturmfels, Gr\ddot obner Bases and Convex Polytopes, Amer. Math. Soc., Providence, RI, 1995.
17. W.V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Springer- Verlag, Berlin, Heidelberg, New York, 1998.