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arxiv: 2604.06084 · v1 · submitted 2026-04-07 · 🧮 math.RT

Relative Serre duality for Coxeter groups

Colton Sandvik This is my paper

Pith reviewed 2026-05-10 18:25 UTC · model grok-4.3

classification 🧮 math.RT
keywords Coxeter groupsSoergel bimodulesparabolic inductionadjointsSerre dualityhomotopy categoriesfinite Coxeter groups
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The pith

Left and right adjoints of parabolic induction are related by the relative full twist for every finite Coxeter group.

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

The paper proves that the left and right adjoints of the parabolic induction functor between homotopy categories of Soergel bimodules for a finite Coxeter group stand in a duality relation given by the relative full twist. This relation had been checked only for symmetric groups, crystallographic groups, and dihedral groups. Establishing the relation in full generality supplies a uniform description of how induction and restriction interact with duality in these categories. A reader would care because the categories organize representations and geometric data attached to Coxeter groups, so the duality gives a concrete way to move between left and right adjunctions.

Core claim

The paper proves the conjecture that the left and right adjoints of the parabolic induction functor between homotopy categories of Soergel bimodules associated to a finite Coxeter group are related by the relative full twist. The proof covers all finite Coxeter groups without exception.

What carries the argument

The relative full twist, which supplies the explicit relation connecting the left adjoint to the right adjoint of parabolic induction.

Load-bearing premise

The specific bimodule constructions and derived-category techniques used in the proof correctly capture the adjoints of parabolic induction for every finite Coxeter group.

What would settle it

A concrete computation in a small non-crystallographic finite Coxeter group, such as H3 or H4, showing that the composition of one adjoint, the relative full twist, and the other adjoint fails to recover the expected identity functor in the homotopy category.

read the original abstract

It was conjectured by Gorsky, Hogancamp, Mellit, and Nakagane that the left and right adjoints of the parabolic induction functor between homotopy categories of Soergel bimodules associated to a finite Coxeter group are related by the relative full twist. Several cases of this conjecture are known including for symmetric groups, crystallographic Coxeter groups, and dihedral groups. We prove this conjecture in complete generality using the theory of Abe-Bott-Samelson bimodules and the Achar-Riche-Vay mixed derived category.

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 paper proves the Gorsky-Hogancamp-Mellit-Nakagane conjecture in complete generality for all finite Coxeter groups: the left and right adjoints of the parabolic induction functor between homotopy categories of Soergel bimodules are related by the relative full twist. The proof identifies these adjoints via Abe-Bott-Samelson bimodules and realizes the relative Serre duality using the Achar-Riche-Vay mixed derived category.

Significance. If the identification holds, the result completes the conjecture beyond the previously known cases (symmetric groups, crystallographic Coxeter groups, dihedral groups) and supplies a uniform framework for adjunctions and duality in Soergel bimodule homotopy categories. The approach leverages two established theories without introducing new parameters or ad-hoc axioms, which strengthens its potential impact on categorification and representation theory of arbitrary finite Coxeter systems.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the assertion that Abe-Bott-Samelson bimodules furnish the correct adjoints of parabolic induction in the homotopy category of Soergel bimodules for arbitrary finite Coxeter groups (including non-crystallographic ones) is load-bearing for the complete-generality claim, yet the manuscript supplies no explicit verification that the bimodule constructions and their adjunction properties survive when the Coxeter matrix is not crystallographic.
  2. [§4] §4 (on the Achar-Riche-Vay mixed derived category): the claim that this category realizes relative Serre duality for every finite Coxeter system requires a concrete check that the duality statements and the identification with the relative full twist remain valid outside the crystallographic setting; without this, the reduction of the conjecture to the cited frameworks is incomplete.
minor comments (2)
  1. Notation for the relative full twist and the parabolic induction functor is introduced without a consolidated table of symbols; adding one would improve readability across sections.
  2. [Introduction] The introduction could briefly recall the precise statement of the Gorsky-Hogancamp-Mellit-Nakagane conjecture (including the precise functors involved) before citing the known cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where the generality of the argument could be made more explicit. We agree that additional clarification will strengthen the paper and will incorporate revisions as detailed below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the assertion that Abe-Bott-Samelson bimodules furnish the correct adjoints of parabolic induction in the homotopy category of Soergel bimodules for arbitrary finite Coxeter groups (including non-crystallographic ones) is load-bearing for the complete-generality claim, yet the manuscript supplies no explicit verification that the bimodule constructions and their adjunction properties survive when the Coxeter matrix is not crystallographic.

    Authors: The Abe-Bott-Samelson bimodules and their adjunction properties with respect to parabolic induction are defined and established in Abe's work for arbitrary Coxeter systems; the constructions rely only on the Coxeter presentation and the general theory of Soergel bimodules, without any crystallographic hypothesis. The manuscript invokes these general results directly. To address the referee's concern, we will add a short clarifying paragraph at the end of §1 that explicitly states the applicability to non-crystallographic groups and points to the relevant statements in Abe's paper. revision: yes

  2. Referee: [§4] §4 (on the Achar-Riche-Vay mixed derived category): the claim that this category realizes relative Serre duality for every finite Coxeter system requires a concrete check that the duality statements and the identification with the relative full twist remain valid outside the crystallographic setting; without this, the reduction of the conjecture to the cited frameworks is incomplete.

    Authors: The Achar-Riche-Vay mixed derived category is built from the Soergel bimodule homotopy category of an arbitrary finite Coxeter group, and the duality statements together with the identification of the relative full twist are established in their framework using only combinatorial data from the Coxeter system. The proofs therein do not invoke root systems or crystallographic conditions. Nevertheless, we will insert a brief remark in §4 that records this observation and confirms that the reduction of the GHMN conjecture proceeds verbatim for non-crystallographic groups. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on independent external frameworks

full rationale

The paper proves the Gorsky-Hogancamp-Mellit-Nakagane conjecture for arbitrary finite Coxeter groups by applying the established theory of Abe-Bott-Samelson bimodules and the Achar-Riche-Vay mixed derived category. These are prior constructions from independent literature (different authors) and are invoked as external tools rather than being redefined or fitted within the paper. No equations reduce by construction to the paper's own inputs, no parameters are fitted to a subset and then relabeled as predictions, and no load-bearing self-citations appear. The central identification of adjoints with the relative full twist is therefore not circular; any questions about applicability to non-crystallographic cases concern correctness or verification, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the standard properties of homotopy categories, adjunctions, and the two named bimodule and derived-category frameworks; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of homotopy categories of Soergel bimodules and adjunctions between them hold for finite Coxeter groups.
    Invoked when defining the parabolic induction functor and its adjoints.
  • domain assumption Abe-Bott-Samelson bimodules and the Achar-Riche-Vay mixed derived category are well-defined and functorial in this setting.
    Used as the technical engine for the general proof.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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