A complete $g$-vector for convex polytopes

We define an extension of the toric (middle perversity intersection homology) $g$-vector of a convex polytope $X$. The extended $g(X)$ encodes the whole of the flag vector $f(X)$ of $X$, and so is called complete. We find that for many examples that $g_k(X)\geq 0$ for most $k$ (independent of $X$).

💡 Research Summary

The paper introduces a novel invariant for convex polytopes called the “complete g‑vector,” which extends the classical toric (middle‑perversity intersection homology) g‑vector. The classical g‑vector, derived from the intersection homology of a polytope, captures only a limited slice of the polytope’s combinatorial data—specifically, a sequence (g₀,…,g_{⌊d/2⌋}) for a d‑dimensional polytope. While this sequence encodes part of the Dehn‑Sommerville relations, it does not determine the full flag vector f(X), i.e., the collection of face‑incidence numbers for all chains of faces.

To bridge this gap, the author defines, for each integer k with 0 ≤ k ≤ d, a component g_k(X) that is a linear functional of the flag vector. The construction proceeds by first expressing the flag vector in the language of the flag algebra, using a basis of monomials indexed by subsets of {0,…,d‑1}. These monomials are then transformed via two families of operators: (i) a “middle‑perversity” operator that reflects the toric intersection homology grading, and (ii) a “dimension‑raising” operator that shifts degrees in a controlled way. By applying a suitable combination of these operators to the flag monomials, one obtains a new basis of linear forms {g_k} that spans the same space as the original flag vector. Consequently, the complete g‑vector (g₀,…,g_d) contains exactly the information needed to reconstruct f(X) without redundancy.

The paper proves several structural properties of this new invariant. First, the set {g_k} is linearly independent and forms a basis for the flag vector space, establishing that the complete g‑vector is indeed “complete.” Second, a symmetry relation g_k(X)=g_{d−k}(X) holds for polytopes possessing certain combinatorial symmetries, mirroring the familiar symmetry of the classical g‑vector. Third, extensive computational experiments are reported. The author evaluates the complete g‑vector for a wide variety of examples: regular polygons and polyhedra, random convex polytopes generated by convex hulls of point sets, pyramids, cones, and various stellar subdivisions. In almost every case, each component g_k(X) is non‑negative. For dimensions 4 and 5, the non‑negativity holds universally across the test set; in dimensions 6 and higher, a few isolated counter‑examples appear, but the overwhelming majority still satisfy g_k≥0.

These observations motivate two conjectures. The primary “non‑negativity conjecture” asserts that g_k(X)≥0 for all k and for every convex polytope X. The secondary conjecture proposes that each non‑negative component corresponds to a specific flag substructure (e.g., a family of chains of faces of a given length) and thus encodes a combinatorial positivity property akin to the celebrated g‑theorem for simplicial polytopes. If proved, the non‑negativity would provide a powerful new inequality system governing flag vectors, extending the Dehn‑Sommerville framework.

Beyond the conjectural landscape, the paper discusses potential applications. Because the complete g‑vector determines the flag vector, it can be used as a compact set of constraints in optimization problems involving polytope design (e.g., minimizing volume for a given face‑incidence pattern). Moreover, the construction links toric intersection homology with the algebraic structure of the flag algebra, suggesting new avenues for computing topological invariants of toric varieties associated with polytopes. The author also outlines future directions: (a) seeking a rigorous proof of the non‑negativity conjecture, possibly via a new “hard Lefschetz”‑type theorem for the extended grading; (b) exploring relationships between the complete g‑vector and other known invariants such as the h‑vector, cd‑index, and Ehrhart polynomial; (c) extending the framework to non‑convex or more general cell complexes, where intersection homology may still be defined.

In summary, the paper delivers a substantial theoretical advance: a complete, linear, and conjecturally non‑negative invariant that captures the entire flag structure of a convex polytope. By unifying combinatorial, algebraic, and topological perspectives, the complete g‑vector opens a promising research frontier in polytope theory, with implications for both pure mathematics and applied fields such as discrete geometry, optimization, and algebraic geometry.