Binomial edge rings of complete bipartite graphs
DOI:
https://doi.org/10.7146/math.scand.a-158136Abstract
We introduce a new class of algebras arising from graphs, called binomial edge rings. Given a graph $G$ on $d$ vertices with $n$ edges, the binomial edge ring of $G$ is defined to be the subalgebra of the polynomial ring with $2d$ variables generated by the binomials which correspond to $n$ edges. In this paper, we calculate a SAGBI basis for this algebra and obtain an initial algebra associated with this SAGBI basis in the case of complete bipartite graphs. It turns out that such an initial algebra is isomorphic to the Hibi ring of a certain poset. Similar phenomenon also occurs in the context of Plücker algebras, so the framework of binomial edge rings can be interpreted as a kind of its generalization.
References
Bruns, W., and Conca, A., SAGBI combinatorics of maximal minors and a SAGBI algorithm, J. Symbolic Comput. 120 (2024), Paper No. 102237, 14pp. https://doi.org/10.1016/j.jsc.2023.102237
Bruns, W., and Gubeladze, J., Polyhedral algebras, arrangements of toric varieties, and their groups, Computational commutative algebra and combinatorics (Osaka, 1999), 1–51, Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, 2002. https://doi.org/10.2969/aspm/03310001
Grayson, D. R., and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at http://www2.macaulay2.com.
Herzog, J., and Hibi, T., Monomial ideals, Graduate Texts in Mathematics, 260. Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-106-6
Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., and Rauh, J., Binomial edge ideals and conditional independence statements, Adv. in Appl. Math. 45 (2010), no. 3, 317–333. https://doi.org/10.1016/j.aam.2010.01.003
Herzog, J., Hibi, T., and Ohsugi, H., Binomial ideals, Graduate Texts in Mathematics, 279. Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-95349-6
Hibi, T., Distributive lattices, affine semigroup rings and algebras with straightening laws, Commutative algebra and combinatorics (Kyoto, 1985), 93–109, Adv. Stud. Pure Math., 11, North-Holland, Amsterdam, 1987. https://doi.org/10.2969/aspm/01110093
Miller, E., and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, 227. Springer-Verlag, New York, 2005. https://doi.org/10.1007/b138602
Ohsugi, H., and Hibi, T., Normal polytopes arising from finite graphs, J. Algebra 207 (1998), no. 2, 409–426 https://doi.org/10.1006/jabr.1998.7476
Ohtani, M., Binomial edge ideals of complete multipartite graphs, Commun. Algebra 41 (2013), no. 10, 3858–3867. https://doi.org/10.1080/00927872.2012.680219
Robbiano, L., and Sweedler, M., Subalgebra bases, Commutative algebra (Salvador, 1988), 61–87, Lecture Notes in Math., 1430, Springer, Berlin, 1990. https://doi.org/10.1007/BFb0085537
Simis, A., Vasconcelos, W. V., and Villarreal, R. H., The integral closure of subrings associated to graphs, J. Algebra 199 (1998), no. 1, 281–289. https://doi.org/10.1006/jabr.1997.7171
Sturmfels, B., Gröbner bases and convex polytopes, University Lecture Series, 8, American Mathematical Society, Providence, RI, 1996. https://doi.org/10.1090/ulect/008
Villarreal, R. H., Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), no. 3, 277–293. https://doi.org/10.1007/BF02568497
