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Gamma positivity, PL homeomorphism types, and orthogonal polynomials

Soohyun Park

Abstract

Using preservations of 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 that are linear in $d$ and those that are superlinear in $d$. 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 with the number of edges and the distance of the $M$-vector condition on $g$-vectors of simplicial spheres from equality. 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 $d$). 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.

Gamma positivity, PL homeomorphism types, and orthogonal polynomials

Abstract

Using preservations of 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 -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 components that are linear in and those that are superlinear in . When the link condition is nontrivial, we show that it gives a lower bound for growth rates of -vector components. This lower bound increases with the number of edges and the distance of the -vector condition on -vectors of simplicial spheres from equality. 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 -vector components with the same index (e.g. -vectors with components increasing quickly compared to ). 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.
Paper Structure (8 sections, 44 theorems, 170 equations)

This paper contains 8 sections, 44 theorems, 170 equations.

Table of Contents

  1. Background
  2. Realizability of nonnegative gamma vectors
  3. General Cohen-Macaulay simplicial complexes and simplicial spheres
  4. Simplicial spheres satisfying the link condition
  5. Main underlying inequalities
  6. Behavior in large dimensions
  7. Summary and asymptotic behavior of individual inequalities
  8. Generalized positivity properties from orthogonal polynomials and lattice paths

Key Result

Proposition 1

(Section linkcondimp) Let $\Delta$ be a $(d - 1)$-dimensional simplicial sphere satisfying the link condition. The global/averaged $M$-vector condition on the $g(\widetilde{\Delta})$ is a lower bound on $\frac{g_{k + 2}(\Delta)}{g_{k + 1}(\Delta)}$ in terms of $\frac{g_{k + 1}(\Delta)}{g_k(\Delta)}$

Theorems & Definitions (98)

  • Proposition 1
  • Theorem 2
  • Proposition 3
  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • ...and 88 more