Hecke combinatorics, K{\aa}hrstr{\"o}m's conditions and Kostant's problem

Samuel Creedon; Volodymyr Mazorchuk

arxiv: 2510.05817 · v2 · submitted 2025-10-07 · 🧮 math.RT

Hecke combinatorics, K{aa}hrstr{\"o}m's conditions and Kostant's problem

Samuel Creedon , Volodymyr Mazorchuk This is my paper

Pith reviewed 2026-05-18 09:33 UTC · model grok-4.3

classification 🧮 math.RT

keywords Hecke algebraKazhdan-Lusztig basisKåhrström conditionsKostant problemcategory Oleft cellscombinatorics

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The pith

Cyclic submodules generated by the dual Kazhdan-Lusztig basis in the regular Hecke module connect to Kåhrström's conditions for Kostant's problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper explores connections between Hecke algebra combinatorics and Kåhrström's conjecture addressing Kostant's problem for simple highest weight modules in the BGG category O for the Lie algebra sl_n. It focuses on cyclic submodules of the regular Hecke module that are generated by elements of the (dual) Kazhdan-Lusztig basis. The authors also examine the question of left cell invariance for Kåhrström's conditions when formulated in both categorical and combinatorial ways. A sympathetic reader would care because these objects could provide a combinatorial handle on determining which modules satisfy Kostant's problem, moving from abstract category theory toward explicit algebraic checks. The work extends existing studies by isolating concrete submodules and invariance questions as central objects of investigation.

Core claim

We study cyclic submodules of the regular Hecke module that are generated by the elements of the (dual) Kazhdan-Lusztig basis as well as the problem of left cell invariance for both categorical and combinatorial Kåhrström's conditions.

What carries the argument

Cyclic submodules of the regular Hecke module generated by (dual) Kazhdan-Lusztig basis elements, together with left cells for Kåhrström's conditions.

If this is right

If left cell invariance holds for the combinatorial conditions, then it is enough to verify Kåhrström's conditions on representatives of each cell rather than on every element.
The generation properties of the cyclic submodules may characterize precisely which weights satisfy the conditions relevant to Kostant's problem.
Combinatorial computations inside the Hecke algebra could decide cases of Kostant's problem that remain open in the categorical setting for sl_n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the combinatorial and categorical versions of the conditions turn out to be equivalent, then Kostant's problem for sl_n reduces to finite checks inside the Hecke algebra.
The same left-cell framework might apply to other semisimple Lie algebras whose Weyl groups admit analogous cell decompositions.
Explicit calculations for small n, such as n=3 or n=4, could be compared against existing lists of modules known to satisfy or fail Kostant's problem.

Load-bearing premise

The combinatorial formulation of Kåhrström's conditions via the Hecke algebra and its cells accurately captures or is equivalent to the categorical conditions arising from the BGG category O for sl_n.

What would settle it

An explicit example of a left cell in the Hecke algebra for some n where the combinatorial version of a Kåhrström condition holds but the corresponding categorical version in category O does not, or vice versa.

read the original abstract

This paper discusses various aspects of the Hecke algebra combinatorics that are related to conditions appearing in K{\aa}hrstr{\"o}m's conjecture that addresses Kostant's problem for simple highest weight modules in the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ for the complex Lie algebra $\mathfrak{sl}_n$. In particular, we study cyclic submodules of the regular Hecke module that are generated by the elements of the (dual) Kazhdan-Lusztig basis as well as the problem of left cell invariance for both categorical and combinatorial K{\aa}hrstr{\"o}m's conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper checks left-cell invariance for Kåhrström conditions in both Hecke combinatorics and category O but does not supply a bridge showing the combinatorial results bear on the categorical annihilator question.

read the letter

The main takeaway is that this work examines cyclic submodules of the regular Hecke module generated by dual Kazhdan-Lusztig basis elements and tests left-cell invariance for Kåhrström's conditions in both the combinatorial Hecke setting and the categorical setting from BGG category O for sl_n. It frames these as aspects related to Kostant's problem but keeps the two sides running in parallel. That is the core of what the paper does. The explicit focus on these submodules and the invariance checks looks like the concrete contribution, and if the authors carry out verifiable calculations for small n or specific cells, that part could be useful to people who already compute with KL bases and cells. Those kinds of checks sometimes turn up patterns that later get used in larger arguments. The soft spot is the missing link. The combinatorial invariance results are interesting on their own terms, yet the paper does not appear to give an isomorphism, functor, or theorem that would let one transfer the combinatorial conclusions to statements about annihilators of simple highest-weight modules in category O. Without that correspondence, it is not clear how far the Hecke-algebra work actually advances the original open question. The assumption that the two formulations are close enough to study side by side is left largely implicit. This paper is written for specialists already working inside Hecke combinatorics and its applications to highest-weight modules. Someone tracking Kåhrström-type conjectures or doing explicit cell computations might pick up a useful example or invariance statement. A broader reader in representation theory would probably find it too narrow unless they have a specific reason to care about these submodules. The calculations and invariance statements look like the sort of incremental, checkable work that can still merit referee attention even if the connection to Kostant's problem stays loose. I would send it to peer review with a request that the authors clarify whether they claim any direct implication from the combinatorial side to the categorical side or whether the two are simply being explored together.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates Hecke algebra combinatorics connected to Kåhrström's conjecture on Kostant's problem for simple highest-weight modules in BGG category O for sl_n. It analyzes cyclic submodules of the regular Hecke module generated by elements of the (dual) Kazhdan-Lusztig basis and studies left-cell invariance under both the combinatorial formulation (via Hecke cells) and the categorical formulation of Kåhrström's conditions.

Significance. If the invariance results and submodule descriptions hold and the two settings are properly linked, the work could supply combinatorial criteria for annihilator questions in category O, extending Kazhdan-Lusztig cell theory to Kostant's problem. The parallel examination of categorical and combinatorial conditions is potentially useful provided an explicit correspondence is established.

major comments (2)

[Introduction] Introduction and abstract: the central claim that combinatorial results on left-cell invariance and cyclic submodules generated by (dual) KL-basis elements bear on Kostant's problem in category O for sl_n requires a precise link (equivalence of conditions or functorial correspondence preserving submodules and invariance); the text treats the settings in parallel without supplying such a bridge, e.g., no explicit isomorphism, natural transformation, or theorem equating combinatorial cell data to categorical annihilator-generation.
[Section on cyclic submodules] Section on cyclic submodules (likely §3 or equivalent): the description of cyclic submodules generated by dual Kazhdan-Lusztig basis elements does not include a statement or proof showing how these submodules map to or determine the annihilator ideals of simple highest-weight modules in BGG category O, which is load-bearing for any application to Kostant's problem.

minor comments (2)

[Abstract] Abstract: the statement of topics studied could be sharpened by indicating the specific theorems or invariance results proved rather than only the areas examined.
Notation: ensure consistent use of dual versus non-dual Kazhdan-Lusztig basis elements when defining the generators of the cyclic submodules.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the connections between the combinatorial and categorical settings. We address the major comments point by point below.

read point-by-point responses

Referee: [Introduction] Introduction and abstract: the central claim that combinatorial results on left-cell invariance and cyclic submodules generated by (dual) KL-basis elements bear on Kostant's problem in category O for sl_n requires a precise link (equivalence of conditions or functorial correspondence preserving submodules and invariance); the text treats the settings in parallel without supplying such a bridge, e.g., no explicit isomorphism, natural transformation, or theorem equating combinatorial cell data to categorical annihilator-generation.

Authors: We agree that the manuscript presents results on left-cell invariance and cyclic submodules in the combinatorial (Hecke-cell) setting and the categorical setting in parallel, without introducing a new explicit isomorphism or natural transformation. The paper relies on the existing Kazhdan-Lusztig correspondence and the formulation of Kåhrström's conjecture to relate the two. In revision we will add a short clarifying subsection in the introduction that recalls the relevant functorial aspects from the literature and states explicitly how the combinatorial invariance properties are expected to inform the categorical annihilator questions under the conjecture. revision: partial
Referee: [Section on cyclic submodules] Section on cyclic submodules (likely §3 or equivalent): the description of cyclic submodules generated by dual Kazhdan-Lusztig basis elements does not include a statement or proof showing how these submodules map to or determine the annihilator ideals of simple highest-weight modules in BGG category O, which is load-bearing for any application to Kostant's problem.

Authors: The section provides a combinatorial description of the cyclic submodules in the regular Hecke module. No direct mapping or proof relating these submodules to annihilator ideals in category O is given, because the manuscript's focus is the Hecke-algebra combinatorics and the invariance statements. We will insert a brief remark in the section noting that, under Kåhrström's conjecture, the generators of the described submodules are expected to correspond to elements whose annihilators are relevant for the simple highest-weight modules, with a reference to the relevant parts of the existing literature on Kostant's problem. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation chain self-contained against external benchmarks

full rationale

The paper examines combinatorial properties of cyclic submodules in the regular Hecke module generated by (dual) Kazhdan-Lusztig basis elements and left-cell invariance under both combinatorial and categorical formulations of Kåhrström's conditions. No equations, fitted parameters, or self-definitional reductions appear in the abstract or described claims. The combinatorial study is presented as parallel to the categorical setting in BGG category O without any quoted step that renames a fit as a prediction or imports uniqueness solely via overlapping-author citation chains. Prior KL-basis and cell results are treated as established external input rather than internally redefined; the work therefore remains non-circular on inspection.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, new entities, or ad-hoc axioms are identifiable from the abstract; the work relies on standard background structures of Hecke algebras, Kazhdan-Lusztig theory, and category O.

axioms (1)

standard math Standard properties of the Kazhdan-Lusztig basis and left cells in Hecke algebras
Invoked when discussing cyclic submodules of the regular Hecke module and left cell invariance.

pith-pipeline@v0.9.0 · 5634 in / 1318 out tokens · 46817 ms · 2026-05-18T09:33:29.417854+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We study cyclic submodules of the regular Hecke module that are generated by the elements of the (dual) Kazhdan-Lusztig basis as well as the problem of left cell invariance for both categorical and combinatorial Kåhrström's conditions.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the combinatorial conditions which appear in Kåhrström's conjecture (where they were formulated for involutions only) are, in fact, left cell invariants.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.