Quadratic initial ideals of root systems
Authors:
Hidefumi Ohsugi and Takayuki Hibi
Journal:
Proc. Amer. Math. Soc. 130 (2002), 19131922
MSC (2000):
Primary 13P10
Published electronically:
December 27, 2001
MathSciNet review:
1896022
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Abstract: Let be one of the root systems , , and and write for the set of positive roots of together with the origin of . Let denote the Laurent polynomial ring over a field and write for the affine semigroup ring which is generated by those monomials with , where if . Let denote the polynomial ring over and write for the toric ideal of . Thus is the kernel of the surjective homomorphism defined by setting for all . In their combinatorial study of hypergeometric functions associated with root systems, Gelfand, Graev and Postnikov discovered a quadratic initial ideal of the toric ideal of . The purpose of the present paper is to show the existence of a reverse lexicographic (squarefree) quadratic initial ideal of the toric ideal of each of , and . It then follows that the convex polytope of the convex hull of each of , and possesses a regular unimodular triangulation arising from a flag complex, and that each of the affine semigroup rings , and is Koszul.
 1.
Annetta
Aramova, Jürgen
Herzog, and Takayuki
Hibi, Finite lattices and lexicographic Gröbner bases,
European J. Combin. 21 (2000), no. 4, 431–439.
MR
1756149 (2001b:06011), http://dx.doi.org/10.1006/eujc.1999.0358
 2.
Jörgen
Backelin and Ralf
Fröberg, Koszul algebras, Veronese subrings and rings with
linear resolutions, Rev. Roumaine Math. Pures Appl.
30 (1985), no. 2, 85–97. MR 789425
(87c:16002)
 3.
Winfried
Bruns, Jürgen
Herzog, and Udo
Vetter, Syzygies and walks, Commutative algebra (Trieste,
1992) World Sci. Publ., River Edge, NJ, 1994, pp. 36–57. MR 1421076
(97f:13024)
 4.
David
Cox, John
Little, and Donal
O’Shea, Ideals, varieties, and algorithms, 2nd ed.,
Undergraduate Texts in Mathematics, SpringerVerlag, New York, 1997. An
introduction to computational algebraic geometry and commutative algebra.
MR
1417938 (97h:13024)
 5.
David
Cox, John
Little, and Donal
O’Shea, Using algebraic geometry, Graduate Texts in
Mathematics, vol. 185, SpringerVerlag, New York, 1998. MR 1639811
(99h:13033)
 6.
W. Fong, Triangulations and Combinatorial Properties of Convex Polytopes, Dissertation, M.I.T., June, 2000.
 7.
Israel
M. Gelfand, Mark
I. Graev, and Alexander
Postnikov, Combinatorics of hypergeometric functions associated
with positive roots, The ArnoldGelfand mathematical seminars,
Birkhäuser Boston, Boston, MA, 1997, pp. 205–221. MR 1429893
(99k:33046)
 8.
James
E. Humphreys, Introduction to Lie algebras and representation
theory, Graduate Texts in Mathematics, vol. 9, SpringerVerlag,
New York, 1978. Second printing, revised. MR 499562
(81b:17007)
 9.
Hidefumi
Ohsugi and Takayuki
Hibi, Compressed polytopes, initial ideals and complete
multipartite graphs, Illinois J. Math. 44 (2000),
no. 2, 391–406. MR 1775328
(2001e:05092)
 10.
Bernd
Sturmfels, Gröbner bases and convex polytopes, University
Lecture Series, vol. 8, American Mathematical Society, Providence, RI,
1996. MR
1363949 (97b:13034)
 1.
 A. Aramova, J. Herzog and T. Hibi, Finite lattices and lexicographic Gröbner bases, Europ. J. Combin. 21 (2000), 431  439. MR 2001b:06011
 2.
 J. Backelin and R. Fröberg, Koszul algebras, Veronese subrings, and rings with linear resolutions, Rev. Roum. Math. Pures Appl. 30 (1985), 85  97. MR 87c:16002
 3.
 W. Bruns, J. Herzog and U. Vetter, Syzygies and walks, in ``Commutative Algebra'' (A. Simis, N. V. Trung and G. Valla, Eds.), World Scientific, Singapore, 1994, pp. 36  57. MR 97f:13024
 4.
 D. Cox, J. Little and D. O'Shea, ``Ideals, Varieties and Algorithms,'' Second Edition, SpringerVerlag, New York, 1996. MR 97h:13024
 5.
 D. Cox, J. Little and D. O'Shea, ``Using Algebraic Geometry,'' SpringerVerlag, New York, 1998. MR 99h:13033
 6.
 W. Fong, Triangulations and Combinatorial Properties of Convex Polytopes, Dissertation, M.I.T., June, 2000.
 7.
 I. M. Gelfand, M. I. Graev and A. Postnikov, Combinatorics of hypergeometric functions associated with positive roots, in ``ArnoldGelfand Mathematics Seminars, Geometry and Singularity Theory'' (V. I. Arnold, I. M. Gelfand, M. Smirnov and V. S. Retakh, Eds.), Birkhäuser, Boston, 1997, pp. 205  221. MR 99k:33046
 8.
 J. E. Humphreys, ``Introduction to Lie Algebras and Representation Theory,'' Second Printing, Revised, SpringerVerlag, Berlin, Heidelberg, New York, 1972. MR 81b:17007
 9.
 H. Ohsugi and T. Hibi, Compressed polytopes, initial ideals and complete multipartite graphs, Illinois J. Math. 44 (2000), 391  406. MR 2001e:05092
 10.
 B. Sturmfels, ``Gröbner Bases and Convex Polytopes,'' Amer. Math. Soc., Providence, RI, 1995. MR 97b:13034
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Additional Information
Hidefumi Ohsugi
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560–0043, Japan
Email:
ohsugi@math.sci.osakau.ac.jp
Takayuki Hibi
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560–0043, Japan
Email:
hibi@math.sci.osakau.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993901064115
PII:
S 00029939(01)064115
Received by editor(s):
August 8, 2000
Received by editor(s) in revised form:
January 29, 2001
Published electronically:
December 27, 2001
Communicated by:
John R. Stembridge
Article copyright:
© Copyright 2001 American Mathematical Society
