The coarse flag Hilbert-Poincaré series of the braid arrangement

TL;DR

The paper analyzes the coarse flag Hilbert-Poincaré series of the braid arrangement by introducing a new ino statistic on pairs $(w,\sigma)\in\mathfrak{S}_{n+1}\times\mathrm{Sym}(n)$, defined via maximal chains in the braid face poset. It proves a precise formula for the numerator $\mathcal{N}_{\mathcal{B}_n}(y,t)$ as $\sum_{w,\sigma} y^{\operatorname{ino}(w,\sigma)} t^{\operatorname{des}(\sigma)}$, thereby providing a companion statistic for descents on the symmetric group. The authors refine this result by introducing a signed edge-labeling $\lambda(w,\sigma)$ and showing $\mathcal{N}_{\mathcal{B}_n}(y,t)=\sum_{w,\sigma} y^{\operatorname{ino}(w,\sigma)} t^{\operatorname{asc}(\lambda(w,\sigma))}$, followed by a local refinement equating ascents of $\lambda$ with descents of $\sigma$ for fixed $w$, and a multivariate framework using $Y=\operatorname{Ino}(w,\sigma)$. A poset construction $P_{w,Y}$ and reverse $(P,\omega)$-partitions underpin the quasisymmetric function arguments, linking to generalized non-left-to-right minima and enabling a robust combinatorial proof. The results illuminate the structure of the numerator and connect to real reflection groups, offering a new tool for analyzing region-descent correspondences and associated invariants in braid-type arrangements.

Abstract

The paper concerns the coarse flag Hilbert-Poincaré series of Maglione-Voll in the case of the braid arrangement associated to the symmetric group. We explicitly construct a companion statistic $\operatorname{ino} : \mathfrak{S}_{n+1} \times \operatorname{Sym}(n) \rightarrow \mathbb{N}$ for the descent statistic on $\operatorname{Sym}(n)$ using reverse $(P,ω)$-partitions and quasisymmetric functions.

Figures (1)

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Theorems & Definitions (30)

• Example 1.2
• Example 1.3
• Theorem 1.4
• Remark 1.5: Motivation
• Proposition 1.6
• Example 1.7
• Theorem 1.8
• Remark 1.9
• Remark 1.10
• Example 2.1