A stability phenomenon in Kazhdan-Lusztig combinatorics

2025/04/08 by Creedon, Samuel, Volodymyr Mazorchuk, Mazorchuk, Volodymyr
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2504.05931

openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, when n goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product \underlineH_x\underlineH_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 O for \mathfraksl_∞ 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 \citeCP.

Citations

• Almost all permutations and involutions are Kostant negative
• Kazhdan-Lusztig left cell preorder and dominance order
• On an infinite limit of BGG categories O
• Indecomposable manipulations with simple modules in category O
• Categories O for Dynkin Borel Subalgebras of Root-Reductive Lie Algebras
• Some homological properties of category O. IV
• Notes on Restricted Inverse Limits of Categories
• Finitary 2-categories associated with dual projection functors
• Cell 2-representations of finitary 2-categories
• Equivalence classes for the mu-coefficient of Kazhdan-Lusztig polynomials in S_n
• Representations of Semisimple Lie Algebras in the BGG Category 𝒪
• Categorification of (induced) cell modules and the rough structure of generalized Verma modules
• Combinatorics of Coxeter Groups
• Kazhdan--Lusztig cells and the Murphy basis
• Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe
• Representations of Coxeter groups and Hecke algebras
• Longest Increasing and Decreasing Subsequences

Related

• A Generalization of Deodhar's Framework for Questions in Kazhdan-Lusztig Theory
• Hecke combinatorics, Kåhrström's conditions and Kostant's problem
• On the action of the long cycle on the Kazhdan-Lusztig basis
• The anti-spherical Hecke categories for Hermitian symmetric pairs
• Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials