Newton Complementary Duals of $f$-Ideals. Issue 2 (15th October 2018)

• Record Type: Journal Article
• Authors: Budd, Samuel; Van Tuyl, Adam
• Is Part Of: Canadian mathematical bulletin =. Volume 62:Issue 2(2019)
• Page Start: 231
• Page End: 241
• Publication Date: 2018-10-15
• Subjects: 05E45, -- 05E40, -- 13F55; f-ideals, -- facet ideal, -- Stanley–Reisner correspondence, -- f-vectors, -- Newton complementary dual, -- Kruskal–Katona; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439518000024
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: A square-free monomial ideal $I$ of $k[x_{1}, \ldots, x_{n}]$ is said to be an $f$ -ideal if the facet complex and non-face complex associated with $I$ have the same $f$ -vector. We show that $I$ is an $f$ -ideal if and only if its Newton complementary dual $\widehat{I}$ is also an $f$ -ideal. Because of this duality, previous results about some classes of $f$ -ideals can be extended to a much larger class of $f$ -ideals. An interesting by-product of our work is an alternative formulation of the Kruskal–Katona theorem for $f$ -vectors of simplicial complexes.