Real toric manifolds associated with chordal nestohedra

TL;DR

This paper provides a combinatorial method to compute the Betti numbers of real toric manifolds linked to chordal nestohedra, using permutation counting derived from poset topology.

Contribution

It introduces an explicit description using alternating B-permutations for chordal building sets, enabling Betti number calculations via permutation enumeration.

Findings

Established EL-shellability of the poset induced by chordal building sets.

Derived a permutation-based formula for Betti numbers of associated real toric manifolds.

Computed Betti numbers for specific cases like real Hochschild varieties.

Abstract

This paper investigates the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We consider the poset topology of a specific poset induced from a chordal building set, and show its EL-shellability. Based on this, we present an explicit description using alternating $\mathcal{B}$-permutations for a chordal building set $\mathcal{B}$, transforming the computing Betti numbers into a counting problem. This approach allows us to compute the $a$-number of a finite simple graph through permutation counting when the graph is chordal. In addition, we provide detailed computations for specific cases such as real Hochschild varieties corresponding to Hochschild polytopes.