Gamma positivity, PL homeomorphism types, and orthogonal polynomials

TL;DR

This paper investigates how topological and combinatorial properties like flagness influence gamma positivity in simplicial spheres, revealing conditions under which gamma vectors grow and connecting these to orthogonal polynomials and lattice path models.

Contribution

It provides a quantitative analysis of the effect of the link condition on gamma positivity and g-vectors, introducing bounds and connections to orthogonal polynomials and lattice paths.

Findings

Link condition has a trivial effect on gamma vectors in high dimensions with nonnegative gamma vectors.

Nontrivial link conditions impose lower bounds on g-vector growth rates.

Gamma positivity relates to orthogonal polynomials and lattice path combinatorics.

Abstract

Using preservations of piecewise linear (PL) homeomorphism types under edge contractions (the link condition) as a topological proxy for flagness, we give a quantitative description of the effect flagness on on gamma positivity of simplicial spheres. In particular, we show that the link condition has a trivial effect on the $g$-vectors (and thus gamma vectors) of high-dimensional simplicial spheres with nonnegative gamma vectors in many cases. Note that this reflects a dichotomy between quantitative behavior arising from $g_1$ components (e.g. measuring ``net number of edge subdivisions'' from the boundary of a cross polytope) that are linear in the dimension and those that are superlinear in the dimension. When the link condition is nontrivial, we show that it gives a lower bound for growth rates of $g$-vector components. This lower bound increases as the number of edges and the distance of the $M$-vector condition on $g$-vectors of simplicial spheres from equality decrease. These lower bounds translate to ones on top gamma vector components and give lower bounds on gamma vector growth rates when the gamma vector components are dominant terms in the $g$-vector components with the same index (e.g. $g$-vectors with components increasing quickly compared to the dimension). Finally, we show that the same results apply to positivity properties generalizing gamma positivity arising from connections between orthogonal polynomials and lattice paths. In the course of doing this, we describe gamma vector components in terms of monomer/dimer covers and point out connections between repeated (stellar) edge subdivisions (Tchebyshev subdivisions) and dimer covers.