The Gamma Dichotomy

The g-vector of a simplicial sphere encodes its combinatorial complexity — the number of faces of each dimension, transformed into a coordinate system that reveals structural properties. Gamma positivity of the g-vector is a strong combinatorial condition: each gamma coefficient is nonneg­ative, implying symmetry and unimodality of the h-vector.

The question: does imposing topological constraints on the simplicial sphere affect gamma positivity?

The link condition — requiring that the link of every edge is a combinatorial sphere — is a topological proxy for flagness, a property central to combinatorial geometry. The paper shows this condition has negligible impact on g-vectors of high-dimensional simplicial spheres with nonneg­ative gamma vectors. Topology doesn’t constrain the combinatorics nearly as much as expected.

A sharper structure appears in the growth rates. The g-vector components exhibit a dichotomy: some grow linearly with dimension, others superlinearly. When the link condition becomes non-trivial — when it actually constrains the simplicial structure — it establishes lower bounds on g-vector component growth. These bounds increase with edge count and with deviation from the M-vector condition, a condition from algebraic geometry.

The connection to orthogonal polynomials extends the framework beyond combinatorics. Repeated edge subdivisions relate to dimer covers (perfect matchings), and the positivity properties transfer through this connection. The same gamma positivity that organizes the combinatorics of simplicial spheres organizes the algebraic structure of lattice paths.

Linear versus superlinear growth. Topological constraints that matter and constraints that don’t. The dichotomy is sharp, and the boundary tells you where topology controls combinatorics.