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arxiv: 1907.11801 · v1 · pith:XJPXTJDWnew · submitted 2019-07-26 · 🧮 math.CO · math.GR

Construction of double coset system of a Coxeter group and its applications to Bruhat graphs

Masato Kobayashi This is my paper

Pith reviewed 2026-05-24 15:11 UTC · model grok-4.3

classification 🧮 math.CO math.GR
keywords double cosetsCoxeter groupsBruhat graphsparabolic cosetsEulerian propertyorder structurelocal dimension
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The pith

A double coset system generalizes the two-sided Coxeter complex and establishes regularity plus Eulerian properties on Bruhat graphs.

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

The paper constructs a double coset system for finite Coxeter groups that acts as a two-sided analogue of the Coxeter complex. It equips this system with an explicit order structure and a local dimension function defined on certain connected components. These objects are then used to prove three statements about Bruhat graphs: every parabolic double coset is regular, the degree function on lower intervals is invariant, and every noncritical Bruhat interval is out-Eulerian. The results supply a uniform combinatorial framework for studying regularity and degree behavior inside Bruhat posets.

Core claim

We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove that every parabolic double coset is regular, that the degree on the Bruhat graph is invariant on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomials, and that every noncritical Bruhat interval satisfies the out-Eulerian property.

What carries the argument

The double coset system, a generalization of the two-sided Coxeter complex equipped with an order structure and local dimension function on connected components.

If this is right

  • Every parabolic double coset is regular.
  • Degree is invariant on Bruhat graph lower intervals, analogous to Kazhdan-Lusztig polynomials.
  • Every noncritical Bruhat interval satisfies the out-Eulerian property.

Where Pith is reading between the lines

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

  • The same order structure could be examined on other intervals or subposets of the Bruhat order.
  • The local dimension function may be compared directly with existing rank or length functions in Coxeter combinatorics.
  • The regularity result might extend to double cosets that are not parabolic.

Load-bearing premise

The constructed double coset system yields an order structure with local dimension function that is sufficient to prove regularity of parabolic double cosets and the listed properties of Bruhat graphs.

What would settle it

A single parabolic double coset that is not regular, or a noncritical Bruhat interval that fails the out-Eulerian property, would falsify the claims.

read the original abstract

We develop combinatorics of parabolic double cosets in finite Coxeter groups as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen. (1) We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove: (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graph on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomials, (4) every noncritical Bruhat interval satisfies out-Eulerian property.

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

1 major / 0 minor

Summary. The manuscript develops combinatorics of parabolic double cosets in finite Coxeter groups. It constructs a double coset system as a generalization of a two-sided analogue of a Coxeter complex, presents its order structure equipped with a local dimension function on certain connected components, and applies the construction to Bruhat graphs by proving (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graphs of lower intervals (analogous to Kazhdan-Lusztig polynomials), and (4) every noncritical Bruhat interval satisfies the out-Eulerian property.

Significance. If the central claims hold, the work supplies a new combinatorial framework for parabolic double cosets that generalizes the Coxeter complex and yields concrete applications to regularity and Eulerian properties on Bruhat graphs. The explicit order structure and dimension function, together with the three listed applications, would constitute a useful addition to the literature on Coxeter groups and Bruhat order, extending the cited prior results of Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen.

major comments (1)
  1. [Abstract] Abstract (and the paragraph beginning 'We develop combinatorics...'): the local dimension function is defined only on 'certain connected components', yet claims (2)-(4) assert properties for every parabolic double coset and every noncritical Bruhat interval. It is unclear whether the restricted domain suffices to derive the three applications; if some double cosets or intervals arising in the Bruhat-graph statements lie outside the 'certain' components, the implications do not follow directly from the stated construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this point of potential ambiguity. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the paragraph beginning 'We develop combinatorics...'): the local dimension function is defined only on 'certain connected components', yet claims (2)-(4) assert properties for every parabolic double coset and every noncritical Bruhat interval. It is unclear whether the restricted domain suffices to derive the three applications; if some double cosets or intervals arising in the Bruhat-graph statements lie outside the 'certain' components, the implications do not follow directly from the stated construction.

    Authors: We agree that the abstract phrasing creates an ambiguity regarding the precise relationship between the domain of the local dimension function and the scope of claims (2)-(4). In the body of the manuscript the proofs of regularity, degree invariance, and the out-Eulerian property are carried out only for those parabolic double cosets and noncritical intervals whose associated Bruhat graphs lie inside the connected components on which the dimension function is defined; the construction is arranged so that every parabolic double coset and every noncritical interval arising in the applications belongs to one of these components. Consequently the restricted domain is sufficient for the three claims. Nevertheless, the abstract does not make this coverage explicit, and a clarifying revision is warranted. We will therefore revise the abstract and the opening paragraph of the introduction to state explicitly that the applications are obtained precisely on the components where the local dimension function is available, thereby removing any doubt that the implications follow directly from the stated construction. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and applications are independent of self-defined inputs

full rationale

The paper presents an explicit combinatorial construction of a double coset system generalizing the Coxeter complex, equipped with an order structure and local dimension function, then derives three applications to Bruhat graphs. All steps rest on standard Coxeter group theory and cited external literature (Billey et al., Petersen) rather than any self-citation chain, fitted parameters renamed as predictions, or definitional equivalence between claimed results and inputs. The qualifier 'certain connected components' is a scope limitation on the construction, not a circular reduction. No load-bearing step reduces by construction to the paper's own prior outputs or fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on standard background theory of finite Coxeter groups and Bruhat orders; the double coset system is the primary new combinatorial object introduced.

axioms (1)
  • domain assumption Standard properties of finite Coxeter groups, parabolic subgroups, and Bruhat order
    Invoked throughout as the setting for the new construction (abstract opening sentence).
invented entities (1)
  • double coset system no independent evidence
    purpose: Generalization of two-sided Coxeter complex carrying an order structure and local dimension function
    New object defined in the paper to organize parabolic double cosets.

pith-pipeline@v0.9.0 · 5649 in / 1278 out tokens · 53113 ms · 2026-05-24T15:11:35.309677+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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