Generalized Poincaré series for models of the braid arrangements (Q1275228)

The braid arrangement \({\mathcal A}_{\ell-1}\) consists of the hyperplanes in \({\mathbb C}^\ell\) defined by the equations \(x_i=x_j\), for \(1\leq i<j\leq \ell\). The complement \({\mathbb C}^\ell - \bigcup {\mathcal A}_{\ell-1}\) is homeomorphic to the space of configurations of \(\ell\) distinct labelled points in the plane. \textit{C. DeConcini} and \textit{C. Procesi} [Sel. Math., New Ser. 1, No. 3, 459-494 (1995; Zbl 0842.14038)] constructed a nonsingular algebraic variety \({\mathcal Y}_{{\mathcal A}_{\ell-1}}\) and an algebraic map \(p: {\mathcal Y}_{{\mathcal A}_{\ell-1}} \to {\mathbb C}^\ell\) such that the preimage of \(\bigcup {\mathcal A}_{\ell-1}\) is a divisor with normal crossings, and \(p\) restricts to an isomorphism on the complement of this divisor. Furthermore, \({\mathcal Y}_{{\mathcal A}_{\ell-1}}\) is homeomorphic to the moduli space of stable \((\ell+1)\)-pointed curves of genus zero. The natural action of \(\Sigma_\ell\) on \({\mathbb C}^\ell\) can be lifted to an action on \({\mathcal Y}_{{\mathcal A}_{\ell-1}}\), yielding a representation of \(\Sigma_\ell\) on the cohomology \(H^*({\mathcal Y}_{{\mathcal A}_{\ell-1}})\). The paper under review gives a complete description of (the character of) this representation. More specifically, the author derives a formula for the generalized Poincaré polynomial \[ P_w(q)=\sum \text{tr} w|_{H^{2i}({\mathcal Y}_{{\mathcal A}_{\ell-1}})}q^i \] for arbitrary \(w\in \Sigma_\ell\). These traces are calculated directly by a combinatorial argument, using a description due to \textit{S. Yuzvinsky} [Invent. Math. 127, No. 2, 319-335 (1997)] of an integral basis for \(H^*({\mathcal Y}_{{\mathcal A}_{\ell-1}})\). Basis elements are indexed by (graph-theoretic) forests with \(\ell\) leaves labelled by the integers \(1, \ldots, \ell\), rooted components, and internal vertices labelled subject to certain valence conditions. The natural \(\Sigma_\ell\) action on these forests, by permutation of the leaf labels, corresponds to the \(\Sigma_\ell\) representation on \(H^*({\mathcal Y}_{{\mathcal A}_{\ell-1}})\). Then the trace of \(w \in \Sigma_\ell\) on \(H^{2i}({\mathcal Y}_{{\mathcal A}_{\ell-1}})\) can be obtained by a counting argument. The count is actually not so easy to carry out, and the resulting formula for \(P_w(q)\) is fairly complicated. In the last part of the paper, the author reformulates the main result in terms of a more general series \(\mathcal H\) in formal variables, which can be described in a more compact form. The Poincaré polynomial \(P_w(q)\) is obtained from \(\mathcal H\) by a certain specialization process involving the cycle structure of \(w\). The expression for \(\mathcal H\) contains certain sums over trees. The author derives closed forms for these sums that may be of independent interest.