A stability phenomenon in Kazhdan-Lusztig combinatorics

TL;DR

The paper proves a stability phenomenon: as the rank grows, the expression of $\hat{\underline{H}}_x\underline{H}_y$ in the dual KL basis stabilizes for $x,y$ in $S_n$, yielding well-defined stabilized coefficients and a stable combinatorial structure. It then leverages this stability to define an action of projective functors on the finite-length subcategory of category $\mathcal{O}$ for $\mathfrak{sl}_\infty$, showing that the restricted limit category inherits a Koszul, well-behaved representation-theoretic framework and that the corresponding Hecke-algebraic data stabilize in the limit. The results connect KL combinatorics, RS correspondence, and $\mu$-function analysis to infinite-rank categorification, with the main payoff being a concrete, stable description of projective functors on $\mathcal{O}^{(\infty,\mathrm{fl})}_0$ and a positive answer to a Koszulity question in this setting. Overall, the work provides a robust bridge between finite KL theory and the representation theory of $\mathfrak{sl}_\infty$, via stable Hecke-algebra actions and restricted-inverse-limit techniques.

Abstract

We prove that, when $n$ goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product $\hat{\underline{H}}_x\underline{H}_y$ of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $S_n$ stabilizes. As an application, we define the action of projective functors on the principal block of category $\mathcal{O}$ for $\mathfrak{sl}_\infty$ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from \cite{CP}.

Theorems & Definitions (18)

• Theorem 1
• Lemma 2
• proof
• Proposition 3
• proof
• Lemma 4
• proof
• Proposition 5
• proof
• Remark 6