Partial Resolutions and Noncrossing Combinatorics

TL;DR

This work develops a unified geometric framework connecting partial Springer resolutions to center elements in Hecke algebras via the Harish-Chandra transform. It establishes explicit formulas for central elements arising from the two partial resolutions, relating pushforwards to relative norm maps and proving a minus-case accessible through Bruhat stratification while the plus-case relies on deep representation-theoretic input (Kawanaka). The authors then apply these results to compute traces, Deodhar-type decompositions of braid/Steinberg varieties, and count points, producing a rich noncrossing Catalan/parking combinatorics that interpolate rational Catalan numbers and parking functions. They further connect these geometric/traces results to HOMFLYPT invariants, Markov traces, and rational Kirkman numbers, elucidating a cohesive picture that ties Lie-theoretic geometry to braid theory and Coxeter combinatorics. The framework yields explicit q-deformations and creates new links among central elements, trace formulas, and combinatorial models such as leaves in associahedra and noncrossing partitions, with potential extensions to other Coxeter types and connections to character varieties.

Abstract

For any finite reductive group, we compute the central elements in its Hecke algebra that arise from partial Springer resolutions via the Harish-Chandra transform. Of the two kinds of partial resolution, the larger is the more interesting case. We deduce formulas for associated Hecke traces, generalizing work of Wan-Wang beyond type $A$, and Deodhar-like decompositions of braid varieties that map to partial Springer resolutions. From the latter, we construct noncrossing sets that interpolate between rational Catalan and parking objects, generalizing our work with Galashin-Lam. In parallel, we establish new formulas for arbitrary $a$-degrees of the HOMFLYPT invariants of positive braid closures, from which we construct noncrossing sets for rational Kirkman numbers.

Figures (1)

• Figure 1: We take $W = S_4$ and $\vec{c} = (s_1, s_2, s_3)$ and $p = 3$. The box for a set $J$ consists of pairs $(v, \vec{\omega})$ such that $\mathsf{Asc}(v) = J$ and $\vec{\omega} \in \mathcal{D}^{(v)}(\vec{c}^p)$. Edges between boxes are inclusions of $J$'s. Each $\vec{\omega}$ is drawn as a $3 \times 3$ array, with elements of $\mathbf{e}_{\vec{\omega}}$ in black. For example, represents $\vec{\omega} = (\mathrm{id}, s_2, s_3, s_1, \mathrm{id}, s_3, s_1, s_2, \mathrm{id})$. In each box, the number to the right, resp. left, of $J$ counts the pairs in the box, resp. among boxes for supersets of $J$. The latter gives $\mathrm{Park}_{W, p}^{J, +}$. The rightmost number in the $(k + 1)$th row is $\mu_4^k(L_{4, 3})|_{\text{\bfseries\itshape{q}} \to 1}$.

Theorems & Definitions (75)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.8
• Corollary 1.9
• Corollary 1.10
• Lemma 2.1