Research

The main decomposition comes from considering doubly Cohen-Macaulay simplicial complexes such that removing a vertex does not change the dimension. In this setting, the coefficients of the h-polynomial are bounded below by those of the double suspension of the link over any edge of the starting simplicial complex. In other words, the remainder has nonnegative coefficients. Note that this is also true if we replace this double suspension by the boundary of a cross polytope of the same dimension by work of Athanasiadis. If we start with a flag sphere, we can “approximate” its h-polynomial by a double suspension of its link over an edge (which itself is a codimension 2 flag sphere) and get to the original flag sphere by a “net nonnegative” number of edge subdivisions. In other words, the second step involves a sequence of edge subdivisions and contractions where there are at least as many edge subdivisions as contractions.

Using this procedure, we can obtain the gamma vector of the flag sphere after taking repeated approximations of the resulting terms by double suspensions of links over edges and taking remainders from a net nonnegative number of edge subdivisions. The vertices that are Boolean (in the sense of a decomposition proposed by Nevo-Petersen-Tenner for simplicial complexes whose f-vectors are equal to the h-vector of a given flag sphere) are from the boundary of a cross polytope and the non-Boolean vertices are those that track remainders from approximations by double suspensions of links over edges. At every step, we have a sum of terms which are products of basis elements of reciprocal/palindromic polynomials of a given dimension and h-polynomials over links over m disjoint edges and points at step m. Note that the degrees involved in the double suspension approximation and remainder parts are formally similar to degrees coming up in geometric Lefschetz decompositions. In particular, we show that remainders of the double suspension approximations are closely related to analogues of the Lefschetz maps which would appear in such a decomposition and their images after an identification involving the Boolean vertices. This involves how the remainders behave under edge subdivisions/contractions and tracing how explicit vertices in the decomposition are involved in spanning the Artinian reduction of the Stanley-Reisner ring of the given flag sphere. Looking towards local vs. global Boolean decompositions (taking “local” to mean links over edges), this indicates that having a change in at most 1 locally Boolean vertex to a non-Boolean one would mean that it extends to a globally Boolean one.

We consider Lefschetz-type decompositions of simplicial complexes whose f-vectors are equal to h-vectors of classes of simplicial complexes closed under (double) suspensions. For the latter, we take those of a given dimension to be PL homeomorphic to each other. When the “primitive” part is Boolean and the simplicial complexes in the class are flag, whether we can patch together such a decomposition from certain codimension 2 local parts depends on if a certain map formally satisfies properties of the Hard Lefschetz theorem. When we consider h-vectors of simplicial pseudomanifolds, these local parts are codimension 2 flag spheres. The map (on the simplicial complexes we take h-vectors of) is a composition of a double suspensions and a “net single edge subdivision”. Finally,  the fact that the simplicial complexes we take f-vectors of can be assumed to be balanced suggests possible algebraic versions of these maps.

The use of the Hard Lefschetz theorem has had a long history of being involved in unimodality results in algebraic combinatorics (e.g. h-vectors of boundaries of simplicial polytopes in work of Stanley). For the most part, they are related to the h-vector, which is naturally related to algebraic structures involving the given simplicial complex. Here, we find f-vectors of (d – 1)-dimensional simplicial complexes with d-colorable 1-skeleta that seem to satisfy a Lefschetz decomposition-like structure with Boolean lattices playing the role of primitive parts. This builds on work of Nevo-Petersen and Nevo-Petersen-Tenner on gamma vectors of odd-dimensional flag spheres. Note that this has some resemblance to a construction of Björner-Frankl-Stanley of balanced shellable simplicial complexes whose h-vectors are equal to f-vectors of with d-colorable 1-skeleta involving “filling in” colors that are missing from individual faces of the starting simplicial complex.

Given a flag sphere, it is known that its h-vector is equal to the f-vector of a simplicial complex with a d-colorable 1-skeleton. We use the unique facet of the latter simplicial complex as a “color palette” for the simplicial complex for the vertex set yielding the Boolean part of the decomposition. This is based on thinking about the 1-dimensional example. The inductive step/local-global argument is based on a partition of unity lemma from Adiprasito’s proof of the g-conjecture by (d – 3)-dimensional subcomplexes whose f-vector is equal to the h-vector of codimension 2 flag spheres covering the original one with the same coloring interpretation. The Dehn-Sommerville relations are involved in the Boolean part of the (d – 2)-dimensional faces. Note that the enumerative consequences of a Boolean decomposition (e.g. what the gamma vector is) can be shown using a *correspondence* with *pairs* of independent components (P, Q) where Q can be any element of the Boolean lattice of a certain size depending on |P|.  The objects  P and Q do not necessarily have to have the same type of object. We can apply our work to positivity questions related to reciprocal/palindromic polynomials associated to flag spheres. Finally, we note that the Lefschetz decomposition we study uses the same degrees as the geometric Lefschetz decomposition for compact Kähler manifolds instead of halved degrees in the usual h-vector setting.

We first study recursive properties of simplicial complexes whose f-vectors attain h-vectors of flag simplicial complexes with natural geometric properties (e.g. Cohen-Macaulay or vertex decomposable). This uses a Boolean decomposition result of Nevo-Petersen-Tenner originally used to study gamma positivity of palindromic polynomials (between unimodality and real-rootedness) to go in the other direction. In the course of doing this, we obtain information on recursions involving these geometric structures and see how these decompositions interact with edge subdivisions. 

Conversely, the focus on f-vectors gives some intrinsic geometric information of gamma vectors from h-polynomials of balanced simplicial spheres (which is known to includes h-polynomials of flag spheres). In particular, we find that their gamma polynomials are closely related to Tchebyshev subdivisions studied by Hetyei (analogous to repeated edge subdivisions) of ``signed unused coloring complexes'' of certain balanced simplicial complexes. Finally, we apply work of Rowlands to show that situations where the gamma vectors are equal to f-vectors of flag simplicial complexes are closely related to instances where f-vectors of cubical complexes built out of Tchebyshev subdivisions of this cell complex are the same as those of some CAT(0) cubical complex.

This started out with attempt to understand the gamma vector of a reciprocal/palindromic polynomial by decomposing it into multiples of gamma vector-like invariants on basis elements spanning the vector space of palindromic/reciprocal polynomials of a given even degree. As a result, we find that the gamma polynomial is essentially an inverted Chebyshev polynomial expansion. An initial consequence is a characterization of real-rootedness of linear combinations of Chebyshev polynomials of the first kind in terms of that of a reciprocal polynomial constructed out of a reverse scaled sequence from the coefficients. This can be taken to be a counterpart to a recent result of Bel-Afia-Meroni-Telen on hyperbolicity of Chebyshev curves.

We can also extract geometric and combinatorial information via this inverted Chebyshev expansion by applying Hetyei’s work on Tchebyshev subdivisions and his earlier work on Tchebyshev posets. In particular, we find that there is a close connection between gamma polynomials and f-vectors of simplicial complexes arising from repeated edge subdivisions taking the type A Coxeter complex to the type B Coxeter complex. Finally, a lift to the ce-indices of posets (modifications of cd-indices) from triangulations of boundaries of cross polytopes can be computed using topological descent statistics. Even without additional assumptions aside from being reciprocal/palindromic, there is a sort of parallel with descent statistics in various gamma positivity examples (e.g. Eulerian polynomials, Chow rings of matroids, h-vectors of nestohedra from work of Foata-Schützenberger, Stump, and Postnikov-Reiner-Williams). 

We give explicit formulas for the gamma vector directly in terms of the input polynomial which extends to arbitrary polynomials. More specifically, we can express them as a linear combination of the coefficients of the starting polynomial (using Catalan numbers and binomial coefficients) or in terms of the derivative of a simple quotient of the input polynomial. The first formula suggests relations to common noncrossing partition/Coxeter group structures in existing gamma positivty examples. The second one shows that the gamma vector measures the difference between local and global information when the input polynomial is the h-polynomial of a simplicial complex. It can also be used to relate signs/inequalities of the gamma vector to upper/lower bounds for the coefficients of the input polynomial. The expression in terms of a derivative and relation to the bounds complements and extends initial observations made by Gal when he defined the gamma vector. Finally, we use the form taken by sums involved while making these estimates and connections between matroid-related examples and intersection numbers to connect these properties of the gamma vector to algebraic structures. For example, this includes characteristic classes involved in log concavity and Schur positivity problems.