Quadratic initial ideals of root systems

Authors:
Hidefumi Ohsugi and Takayuki Hibi

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1913-1922

MSC (2000):
Primary 13P10

DOI:
https://doi.org/10.1090/S0002-9939-01-06411-5

Published electronically:
December 27, 2001

MathSciNet review:
1896022

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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**, https://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****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****4.**David Cox, John Little, and Donal O’Shea,*Ideals, varieties, and algorithms*, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. An introduction to computational algebraic geometry and commutative algebra. MR**1417938****5.**David Cox, John Little, and Donal O’Shea,*Using algebraic geometry*, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR**1639811****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 Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 205–221. MR**1429893**, https://doi.org/10.1007/978-1-4612-4122-5_10**8.**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR**499562****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****10.**Bernd Sturmfels,*Gröbner bases and convex polytopes*, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR**1363949**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
13P10

Retrieve articles in all journals with MSC (2000): 13P10

Additional Information

**Hidefumi Ohsugi**

Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560–0043, Japan

Email:
ohsugi@math.sci.osaka-u.ac.jp

**Takayuki Hibi**

Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560–0043, Japan

Email:
hibi@math.sci.osaka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-01-06411-5

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