Introduces Chow functions on posets that parallel KLS functions and relate positivity conjectures on face lattices, matroid Chow rings, and Bruhat intervals.
Combinatorial
5 Pith papers cite this work. Polarity classification is still indexing.
5
Pith papers citing it
abstract
Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we introduce a version of the K\"ahler package beyond positivity, allowing us to prove the hard Lefschetz theorem for toric varieties (and beyond) even when the ample cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a non-degeneracy relation at torus-invariant subspaces, allowing us to state and prove a generalization of theorems of Hall and Laman in the setting of toric varieties and, more generally, the face rings of Hochster, Reisner and Stanley. This has several applications: - We fully characterize the possible face numbers of simplicial rational homology spheres, resolving the $g$-conjecture of McMullen in full generality and generalizing Stanley's earlier proof for simplicial polytopes. The same methods also verify a conjecture of K\"uhnel: if $M$ is a triangulated closed $(d-1)$-manifold on $n$ vertices, then \[\binom{d+1}{j}\mathrm{b}_{j-1}(M)\ \le \ \binom{n-d+j-2}{j}\ \quad \text{for}\ 1\le j\le \frac{d}{2}.\] - We prove that for a simplicial complex that embeds into $\mathbb{R}^{2d}$, the number of $d$-dimensional simplices exceeds the number of $(d-1)$-dimensional simplices by a factor of at most $d+2$. This generalizes a result going back to Descartes and Euler, and resolves the Gr\"unbaum-Kalai-Sarkaria conjecture. We obtain from this a generalization of the celebrated crossing lemma: For a map of a simplicial complex $\Delta$ into $\mathbb{R}^{2d}$, the number of pairwise intersections of $d$-simplices is at least \[\frac{f_d^{d+2}(\Delta)}{(d+3)^{d+2}f_{d-1}^{d+1}(\Delta)}\] provided $f_d(\Delta)> (d+3)f_{d-1}(\Delta)$.
citation-role summary
method 1
citation-polarity summary
verdicts
UNVERDICTED 5roles
method 1polarities
use method 1representative citing papers
Frobenius identities for the volume map on Cohen-Macaulay rings give sufficient conditions for anisotropy and Hard Lefschetz in Gorenstein quotients and deduce the g-theorem for simplicial spheres plus the Ohsugi-Hibi conjecture.
Iterated power set applications generate a hierarchy of operads linking the permutative operad to triassociative, substitution, and composition operads, plus a new operad on relative simplicial complexes governed by join polyhedral products.
Simplicial spheres without large missing faces satisfy g-number lower bounds in terms of graph independence numbers, including g2 ≥ (1/2 − δ(d))f0 for flag spheres with δ(d) → 0 as d → ∞.
Proves gamma-positivity for Hilbert-Poincaré polynomials of Chow rings of matroids with complete and flag building sets, yielding combinatorial analogues of classical positivity conjectures and an explicit simplicial complex realizing the gamma-vector.
citing papers explorer
-
Chow functions for partially ordered sets
Introduces Chow functions on posets that parallel KLS functions and relate positivity conjectures on face lattices, matroid Chow rings, and Bruhat intervals.
-
Frobenius identities for the volume map on Cohen--Macaulay rings
Frobenius identities for the volume map on Cohen-Macaulay rings give sufficient conditions for anisotropy and Hard Lefschetz in Gorenstein quotients and deduce the g-theorem for simplicial spheres plus the Ohsugi-Hibi conjecture.
-
Power set operads
Iterated power set applications generate a hierarchy of operads linking the permutative operad to triassociative, substitution, and composition operads, plus a new operad on relative simplicial complexes governed by join polyhedral products.
-
Lower bounds on the $g$-numbers of spheres without large missing faces
Simplicial spheres without large missing faces satisfy g-number lower bounds in terms of graph independence numbers, including g2 ≥ (1/2 − δ(d))f0 for flag spheres with δ(d) → 0 as d → ∞.
-
Matroid analogues of Gal's conjecture
Proves gamma-positivity for Hilbert-Poincaré polynomials of Chow rings of matroids with complete and flag building sets, yielding combinatorial analogues of classical positivity conjectures and an explicit simplicial complex realizing the gamma-vector.