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arxiv: 2410.22599 · v6 · submitted 2024-10-29 · 🧮 math.GR · math.CO

Combinatorics of cone types in Coxeter groups

Yeeka Yau This is my paper

Pith reviewed 2026-05-23 19:14 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords Coxeter groupscone typesinversion setsweak orderconvexitycombinatorics of groups
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The pith

In a Coxeter group, the elements y that share exactly one inversion β with a fixed x form a convex set in the weak order with a unique minimal representative.

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

The paper proves that for any element x and any root β in its inversion set Φ(x), the set of all y such that Φ(x) ∩ Φ(y) equals exactly {β} is convex in the weak order on W. This set also has a unique smallest element under that order. The result ties directly to the definition of cone types, which classify elements according to the combinatorial shape of their inversion sets. Because the sets are convex and have minimal representatives, one can decide whether two elements have the same cone type by comparing their minimal representatives rather than checking every possible intersection.

Core claim

We prove that for any element x in a Coxeter group W and root β in its inversion set Φ(x), the set of elements y ∈ W satisfying Φ(x) ∩ Φ(y) = {β} is convex in the weak order and admits a unique minimal representative. This combinatorial fact is directly connected to the determination of cone types and yields efficient methods for checking when two elements of W belong to the same cone type.

What carries the argument

The intersection condition Φ(x) ∩ Φ(y) = {β} that isolates the elements sharing exactly one inversion with x, shown to be convex with a unique minimal element in the weak order.

If this is right

  • Cone types of elements can be computed by locating the unique minimal representative in each such intersection set.
  • Equality of cone types between two elements reduces to checking whether their minimal representatives coincide.
  • The convexity supplies a structural reason why cone-type partitions behave regularly under the weak order.

Where Pith is reading between the lines

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

  • The minimal representatives could serve as canonical labels when enumerating the distinct cone types that appear in a given Coxeter group.
  • The same intersection sets might be used to define a partial order or filtration on the set of all cone types.

Load-bearing premise

The standard definitions and basic properties of inversion sets, the weak order, and cone types hold for an arbitrary Coxeter group without extra restrictions.

What would settle it

A concrete counterexample consisting of a specific Coxeter group W, an element x, and a root β where the set {y | Φ(x) ∩ Φ(y) = {β}} fails to be convex or has more than one minimal element under the weak order.

Figures

Figures reproduced from arXiv: 2410.22599 by Yeeka Yau.

Figure 1
Figure 1. Figure 1: Let x and y be the elements represented by the blue and red dots respectively. The cone type T := T(x −1 ) = T(y −1 ) is represented by the gray shaded region. Let β ∈ ∂T correspond to the blue hyperplane. The black hyperplanes correspond to the remaining boundary roots of T. The region bounded by the dotted lines is Q(T). The region bounded by the red highlighted lines is ∂T(y −1 )β and the region bounded… view at source ↗
Figure 2
Figure 2. Figure 2: Let the element x be represented by the alcove with the black dot. The grey shaded region is the cone type T := T(x −1 ) (the darker alcove represents the identity). The elements w in the yellow shaded region are the elements such that T(w −1 ) = T (i.e. the cone type part Q(T) of T corresponding to T). The remaining coloured shaded regions are the sets ∂T(x −1 )β for each β ∈ ∂T and the corresponding colo… view at source ↗
Figure 3
Figure 3. Figure 3: The cone type partition T for the Coxeter group of type Ge2. For each cone type T, the part Q(T) contains the elements x such that T(x −1 ) = T. The shaded in alcove of each part represents the inverse of the unique minimal length cone type representatives m−1 T . These are the gates of T . Theorem 2.2. [21, Theorem 1] For each cone type T there is a unique element mT ∈ W of minimal length such that T(mT )… view at source ↗
Figure 4
Figure 4. Figure 4: The partition T 0 = T for the rank 3 Coxeter groups of affine type. The green alcoves represent the tight gates and the cyan alcoves are the gates which are non-trivial joins of tight gates [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Data for low rank affine and compact hyperbolic Coxeter groups. 7. Ultra-Low Elements The remainder of this paper is devoted to proving Theorem 5. Theorem 6 is a consequence of the compu￾tations in this section and Section 8. 7.1. Dihedral groups, right-angled and complete graph Coxeter groups. We begin with some basic observations and useful results regarding inversion sets of finite Coxeter groups and di… view at source ↗
Figure 6
Figure 6. Figure 6: Throughout this chapter let the simple reflections s, t, u correspond to the vertices reading left to right and denote a = ms,t and b = mt,u. The corresponding cone type automata is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The automaton Ao for (W, S) of Type I with b odd (the illustration includes all states for the case b = 7 with the trailing blue and black dots indicating the additional ele￾ments of W⟨t,u⟩ when b > 7). The node coloured black denotes the identity (start state) and the red, black and blue arrows correspond to s, t and u transitions respectively. Transitions into gray filled nodes informs the reader to cont… view at source ↗
Figure 8
Figure 8. Figure 8: The automaton Ao for (W, S) of Type II with a and b odd (the illustration includes all states for the case a = b = 5 with the trailing dashes indicating additional elements of W⟨t,u⟩ and W⟨t,s⟩ when a, b > 5). One may note that the automaton here is similar to Type I but less complicated in the sense that there are less nodes at the top of the figure. Again, the states highlighted green indicate the (inver… view at source ↗
Figure 9
Figure 9. Figure 9: The automaton Ao for (W, S) of Type III. All transitions and ultra-low elements remain the same as in type II, with the exception of the states at the "top" of the diagram (compare the top part of [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
read the original abstract

In this article, we establish some new combinatorial properties of cone types in Coxeter groups. Firstly, we show that for any element $x$ in a Coxeter group $W$ and root $\beta$ in its inversion set $\Phi(x)$, the set of elements $y \in W$ satisfying $\Phi(x) \cap \Phi(y) = \{ \beta \}$ is convex in the weak order and admits a unique minimal representative. This is strongly connected to determining the cone type of elements of $W$ and leads to efficient computational methods to determine whether arbitrary elements of $W$ have the same cone type.

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

0 major / 2 minor

Summary. The manuscript establishes new combinatorial properties of cone types in Coxeter groups. It proves that for any element x in a Coxeter group W and root β in its inversion set Φ(x), the set of y ∈ W with Φ(x) ∩ Φ(y) = {β} is convex in the weak order and admits a unique minimal representative. This is applied to determining cone types of elements and yields efficient computational methods for checking whether arbitrary elements share the same cone type.

Significance. If the central claims hold, the convexity and uniqueness results supply a new structural tool for analyzing inversion sets and weak order in arbitrary Coxeter groups (finite or infinite). The connection to cone types and the resulting algorithms constitute a concrete advance in the combinatorial study of reflection groups, with potential utility for computational enumeration and classification tasks.

minor comments (2)
  1. The abstract states the main theorem but does not indicate the proof strategy or key lemmas; a one-sentence outline of the argument would improve readability for readers outside the immediate subfield.
  2. Notation for cone types is introduced without an explicit forward reference to the section where the definition is recalled or extended; adding a parenthetical pointer would clarify the logical flow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised, so we have no points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct combinatorial theorem on inversion sets Φ(x) and convexity in the weak order for arbitrary Coxeter groups W, using only the standard definitions of these objects. No parameters are fitted, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definition. The central claim is an assertion proved from the given axioms of Coxeter groups and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a proof within the established theory of Coxeter groups; the abstract introduces no free parameters, new entities, or ad-hoc axioms beyond the standard domain assumptions of the field.

axioms (1)
  • domain assumption Standard definitions and properties of Coxeter groups, root systems, inversion sets Φ(x), weak order, and cone types hold for arbitrary W.
    The convexity statement is formulated inside this framework.

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Reference graph

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