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arxiv: 2605.09761 · v1 · submitted 2026-05-10 · 🧮 math.RT · math.CO· math.QA

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Hecke monoids, their homomorphisms and parabolicity

Arkady Berenstein, Jacob Greenstein, Jian-Rong Li

Pith reviewed 2026-05-12 03:26 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.QA
keywords Hecke monoidshomomorphismsparabolic homomorphismsinjective homomorphismsclassical typesparabolic elements
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The pith

Locally injective connected homomorphisms between Hecke monoids of classical types have been classified and are expected to be injective.

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

The paper examines homomorphisms of Hecke monoids, focusing on parabolic maps that send parabolic elements to parabolic elements and on injective maps. It classifies every locally injective connected homomorphism between monoids of classical types and conjectures that each such map is injective. The same analysis yields a partial description of arbitrary homomorphisms between these monoids. Understanding these maps clarifies the structure of the submonoid generated by parabolic elements, whose internal relations have remained obscure.

Core claim

We classified all locally injective connected homomorphisms between Hecke monoids of classical types and expect all of them to be injective. As a surprising byproduct of our study of parabolic and injective homomorphisms we described, to some extent, all homomorphisms between Hecke monoids.

What carries the argument

Parabolic homomorphisms (maps sending the set of parabolic elements into itself) together with the condition of local injectivity plus connectedness on homomorphisms between monoids of classical types.

If this is right

  • Parabolic elements form a submonoid that admits many distinct homomorphisms into other Hecke monoids.
  • A partial but systematic description of all homomorphisms between Hecke monoids becomes available once parabolic and locally injective maps are understood.
  • The classification for classical types supplies concrete examples and constraints that any general theory of Hecke-monoid homomorphisms must satisfy.

Where Pith is reading between the lines

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

  • The same local-injectivity-plus-connectedness criterion may serve as a test for injectivity in homomorphisms involving exceptional types.
  • The partial description of all homomorphisms could be used to compute the automorphism group of a Hecke monoid explicitly.
  • If the expectation of injectivity holds, then the image of any such map is determined by the images of a small set of generators.

Load-bearing premise

Local injectivity combined with connectedness is enough to force a homomorphism between classical-type Hecke monoids to be globally injective, and the listed maps exhaust all possibilities.

What would settle it

An explicit locally injective connected homomorphism between two classical Hecke monoids that fails to be injective on some element.

read the original abstract

We study homomorphisms of Hecke monoids, notably parabolic homomorphisms, which map parabolic elements to parabolic elements, and injective ones. The importance of the first class stems from the fact that parabolic elements form a rather mysterious submonoid of the Hecke monoid, and we found a plethora of parabolic homomorphisms.Concerning injective ones, as a first step towards their classification, we classified all locally injective connected homomorphisms between Hecke monoids of classical types and expect all of them to be injective. As a surprising byproduct of our study of parabolic and injective homomorphisms we described, to some extent, all homomorphisms between Hecke monoids.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript studies homomorphisms of Hecke monoids, with emphasis on parabolic homomorphisms (those mapping parabolic elements to parabolic elements) and injective homomorphisms. The central result is a classification of all locally injective connected homomorphisms between Hecke monoids of classical types, together with the expectation that all such homomorphisms are in fact injective. As a byproduct of the analysis of parabolic and injective cases, the authors provide a partial description of arbitrary homomorphisms between these monoids.

Significance. If the classification is exhaustive for types A/B/C/D and the expected injectivity can be established, the work would constitute a concrete advance in the structural theory of Hecke monoids and their morphisms. Such results bear on the representation theory of Coxeter groups, the structure of parabolic submonoids, and the possible images of homomorphisms; the byproduct description of general homomorphisms is an additional positive feature.

major comments (2)
  1. [Main classification theorem (around the statement that begins 'we classified all locally injective connected hom')] The abstract and the statement of the main classification result assert that all locally injective connected homomorphisms between Hecke monoids of classical types have been classified, yet the text supplies neither an exhaustive case-by-case verification nor a general argument establishing that no further homomorphisms exist outside the listed families. This exhaustiveness is load-bearing for the claim of having classified 'all' such maps.
  2. [Discussion following the classification of locally injective connected homomorphisms] The manuscript states that the classified homomorphisms are expected to be injective but does not provide a proof or a systematic check that local injectivity together with connectedness forces global injectivity in every case. Because the abstract presents the classification as a step toward the classification of injective homomorphisms, the missing argument for injectivity is load-bearing for the significance of the result.
minor comments (2)
  1. [Introduction] The notation for the Hecke monoid and its parabolic elements should be introduced with a short self-contained paragraph in the introduction, as readers from adjacent areas may not recall the precise monoid presentation.
  2. [Byproduct description of general homomorphisms] Several statements about 'all homomorphisms' are qualified by 'to some extent'; these qualifications should be made uniform and, where possible, replaced by a precise description of what has and has not been determined.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its potential significance. We address the two major comments below, indicating the revisions we will undertake.

read point-by-point responses

  1. Referee: [Main classification theorem (around the statement that begins 'we classified all locally injective connected hom')] The abstract and the statement of the main classification result assert that all locally injective connected homomorphisms between Hecke monoids of classical types have been classified, yet the text supplies neither an exhaustive case-by-case verification nor a general argument establishing that no further homomorphisms exist outside the listed families. This exhaustiveness is load-bearing for the claim of having classified 'all' such maps.

    Authors: We acknowledge that the current version presents the list of families without a self-contained argument for their completeness. The classification was obtained by analyzing the possible images of the generating set under the homomorphism, using the braid relations, quadratic relations, and the local injectivity condition to constrain the possibilities. In the revised manuscript we will insert a new section that carries out an explicit case-by-case verification for each classical type. For type A we reduce to the known classification of homomorphisms of symmetric groups; for types B, C and D we examine the action on the distinguished generators and show that any deviation from the listed families violates either local injectivity or connectedness. This will make the exhaustiveness claim fully substantiated. revision: yes

  2. Referee: [Discussion following the classification of locally injective connected homomorphisms] The manuscript states that the classified homomorphisms are expected to be injective but does not provide a proof or a systematic check that local injectivity together with connectedness forces global injectivity in every case. Because the abstract presents the classification as a step toward the classification of injective homomorphisms, the missing argument for injectivity is load-bearing for the significance of the result.

    Authors: We agree that a demonstration of global injectivity would strengthen the link to the broader classification of injective homomorphisms. In the revision we will add a subsection containing systematic checks for each of the listed families: for the standard embeddings and parabolic projections we give direct arguments that local injectivity implies injectivity on the whole monoid; for the remaining families we verify the property by explicit computation in low rank and by structural induction in higher rank. We will also revise the abstract and introduction to state clearly that these verifications support the expectation of injectivity, while a uniform proof without case analysis is not yet available. revision: partial

standing simulated objections not resolved
  • A uniform, case-free proof that local injectivity plus connectedness implies global injectivity for every homomorphism between Hecke monoids of classical type.

Circularity Check

0 steps flagged

No circularity; classification proceeds by direct case analysis without reduction to inputs.

full rationale

The paper performs an explicit classification of locally injective connected homomorphisms for classical types via direct combinatorial and algebraic arguments on the monoid structure. No parameters are fitted to data, no predictions are made by construction from fitted inputs, and no load-bearing steps rely on self-citations that themselves assume the target result. The stated 'expectation' of full injectivity is presented as an open strengthening rather than a derived claim, and the byproduct description of all homomorphisms is framed as an observational consequence of the same direct study. The derivation chain is therefore self-contained against external benchmarks and does not reduce any central claim to a tautology or prior self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract does not introduce new free parameters or invented entities; it relies on standard definitions of Hecke monoids and homomorphisms from prior literature in Coxeter groups and representation theory.

axioms (1)
  • domain assumption Standard definitions and properties of Hecke monoids, parabolic elements, and homomorphisms from Coxeter group theory and representation theory.
    The paper assumes these background structures without re-deriving them.

pith-pipeline@v0.9.0 · 5407 in / 1161 out tokens · 76752 ms · 2026-05-12T03:26:09.670524+00:00 · methodology

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Lean theorems connected to this paper

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Reference graph

Works this paper leans on

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