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

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.**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, Springer-Verlag, New York, 1996. MR**97h:13024****5.**D. Cox, J. Little and D. O'Shea, ``Using Algebraic Geometry,'' Springer-Verlag, 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*``Arnold-Gelfand 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, Springer-Verlag, 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**

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