Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T22:27:22.095Z Has data issue: false hasContentIssue false
  • English
  • Français

Centrally Symmetric Configurations of Integer Matrices

Published online by Cambridge University Press:  11 January 2016

Hidefumi Ohsugi  and
Takayuki Hibi
Hidefumi Ohsugi
Affiliation:
Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, 669-1337, Japan, [email protected]
Takayuki Hibi
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan, [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.

The concept of centrally symmetric configurations of integer matrices is introduced. We study the problem when the toric ring of a centrally symmetric configuration is normal and when it is Gorenstein. In addition, Gröbner bases of toric ideals of centrally symmetric configurations are discussed. Special attention is given to centrally symmetric configurations of unimodular matrices and to those of incidence matrices of finite graphs.

Keywords

13P1052B20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 216 , 2014 , pp. 153 - 170
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Ardila, F., Beck, M., Hoşten, S., Pfeifle, J., and Seashore, K., Root polytopes and growth series of root lattices, SIAM J. Discrete Math. 25 (2011), 360–378. MR 2801233. DOI 10.1137/090749293.Google Scholar
[2] CoCoATeam, CoCoA: A System for Doing Computations in Commutative Algebra, http://cocoa.dima.unige.it (accessed 29 December 2014).Google Scholar
[3] De Negri, E. and Hibi, T., Gorenstein algebras of Veronese type, J. Algebra 193 (1997), 629–639. MR 1458806. DOI 10.1006/jabr.1997.6990.Google Scholar
[4] Grossman, J. W., Kulkarni, D. M., and Schochetman, I. E., On the minors of an incidence matrix and its Smith normal form, Linear Algebra Appl. 218 (1995), 213–224. MR 1324059. DOI 10.1016/0024-3795(93)00173-W.CrossRefGoogle Scholar
[5] Hibi, T., Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, Australia, 1992. MR 3183743.Google Scholar
[6] Ohsugi, H., A geometric definition of combinatorial pure subrings and Gröbner bases of toric ideals of positive roots, Comment. Math. Univ. St. Pauli 56 (2007), 27–44. MR 2356748.Google Scholar
[7] Ohsugi, H., Herzog, J., and Hibi, T., Combinatorial pure subrings, Osaka J. Math. 37 (2000), 745–757. MR 1789447.Google Scholar
[8] Ohsugi, H. and Hibi, T., Normal polytopes arising from finite graphs, J. Algebra 207 (1998), 409–426. MR 1644250. DOI 10.1006/jabr.1998.7476.CrossRefGoogle Scholar
[9] Ohsugi, H. and Hibi, T., Koszul bipartite graphs, Adv. in Appl. Math. 22 (1999), 25–28. MR 1657721. DOI 10.1006/aama.1998.0615.CrossRefGoogle Scholar
[10] Ohsugi, H. and Hibi, T., Toric ideals generated by quadratic binomials, J. Algebra 218 (1999), 509–527. MR 1705794. DOI 10.1006/jabr.1999.7918.CrossRefGoogle Scholar
[11] Schrijver, A., Theory of Linear and Integer Programming, Wiley, Chichester, 1986. MR 0874114.Google Scholar
[12] Sturmfels, B., Gröbner Bases and Convex Polytopes, Univ. Lecture Ser. 8, Amer. Math. Soc., Providence, 1996. MR 1363949.Google Scholar