[Paper Review] The local $h$-vector of the cluster subdivision of a simplex

Christos A. Athanasiadis, Christina Savvidou|arXiv (Cornell University)|Apr 2, 2012

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper computes the local $h$-vector of the cluster subdivision of a simplex associated with a finite root system $\Phi$, showing its $\gamma$-vector is nonnegative. It provides combinatorial interpretations in terms of noncrossing partitions for types $A$ and $B$, and establishes analogous results for the barycentric subdivision using permutation statistics and the Foata-Sch"utzenberger-Strehl action.

ABSTRACT

The cluster complex $Δ(Φ)$ is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system $Φ$. The positive part of $Δ(Φ)$ naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of $Φ$. The local $h$-vector of this subdivision, in the sense of Stanley, is computed and the corresponding $γ$-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local $h$-vector and the corresponding $γ$-vector are provided for the classical root systems, in terms of noncrossing partitions of types $A$ and $B$. An analogous result is given for the barycentric subdivision of a simplex.

Motivation & Objective

• To compute the local $h$-vector of the cluster subdivision of a simplex arising from a finite root system $\Phi$.
• To provide combinatorial interpretations of the local $h$-vector and its $\gamma$-vector for classical root systems using noncrossing partitions.
• To extend the analysis to the barycentric subdivision of a simplex and relate its local $h$-vector to permutation statistics.
• To demonstrate the nonnegativity of the $\gamma$-vector for cluster subdivisions, supporting a broader conjecture on flag geometric subdivisions.
• To unify combinatorial structures—noncrossing partitions and derangements—via the Foata-Sch"utzenberger-Strehl action on permutations.

Proposed method

• Define the cluster subdivision $\Gamma(\Phi)$ as the positive part of the cluster complex $\Delta(\Phi)$, which geometrically subdivides the simplex on the simple roots.
• Use Stanley's notion of the local $h$-polynomial $\ell_V(\Gamma,x)$ as an alternating sum over face restrictions of the subdivision.
• Express the local $h$-polynomial in $\gamma$-basis via $\ell_V(\Gamma,x) = \sum_{i=0}^{\lfloor n/2\rfloor} \xi_i x^i (1+x)^{n-2i}$, defining the local $\gamma$-polynomial.
• Establish combinatorial interpretations of $\xi_i$ using permutations in $\mathcal{E}_n$ (excedance-avoiding) and $\mathcal{D}_n$ (derangements), with equivalence classes under the Foata-Sch"utzenberger-Strehl action.
• Leverage the involution $\psi_i$ to define equivalence classes and derive generating functions of the form $x^{\text{des}(w)}(1+x)^{n-2\text{des}(w)}$.
• Prove that $\xi_i$ counts permutations with no double descent and $i$ descents, or derangements with $n-i$ excedances and no specific index pattern.

Experimental results

Research questions

• RQ1 What is the explicit form of the local $h$-vector for the cluster subdivision of a simplex associated with a finite root system $\Phi$?
• RQ2 How can the entries of the local $h$-vector and its $\gamma$-vector be interpreted combinatorially for types $A$ and $B$?
• RQ3 What is the relationship between the local $h$-vector of the barycentric subdivision and permutation statistics such as excedances and descents?
• RQ4 Does the local $\gamma$-vector of the cluster subdivision have nonnegative coefficients, and can this be proven combinatorially?
• RQ5 Can the local $\gamma$-vector be uniformly interpreted across different root systems, particularly in types $A_n$ and $B_n$?

Key findings

• The local $\gamma$-vector of the cluster subdivision of a simplex is nonnegative for all finite root systems, including types $A_n$ and $B_n$.
• For type $A_n$, the $i$-th entry $\xi_i$ of the local $\gamma$-vector counts the number of permutations in $\mathcal{E}_n$ with no double descent and exactly $i$ descents.
• For type $B_n$, the local $\gamma$-vector entries are interpreted via noncrossing partitions of type $B$, generalizing the type $A$ case.
• The local $h$-polynomial of the barycentric subdivision equals the generating function of excedances over derangements, $\sum_{w \in \mathcal{D}_n} x^{\text{ex}(w)}$.
• The local $\gamma$-vector of the barycentric subdivision counts permutations with no double descent and $i$ descents, matching the type $A$ cluster case.
• The local $h$-polynomial of the cluster subdivision satisfies $\ell_V(\Gamma,x) = \sum_{u \in \mathcal{E}_n} x^{\text{des}(u)}$, linking it to descent statistics via the Foata-Sch"utzenberger-Strehl action.