arxiv: 2604.22972 · v1 · submitted 2026-04-24 · 🧮 math.RT · math.CO

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The coloured mutation class of mathbb{D}_n- quivers and their application to m-cluster tilted algebras

Claudio Qureshi, Viviana Gubitosi

Pith reviewed 2026-05-08 09:03 UTC · model grok-4.3

classification 🧮 math.RT math.CO

keywords m-coloured quiversquiver mutationD_n quiversm-cluster-tilted algebrasGabriel quiversDynkin typecluster algebrasrepresentation theory

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The pith

An explicit combinatorial characterization identifies every m-coloured quiver reachable by m-coloured mutations from a type D_n quiver.

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

The authors give a complete list of combinatorial rules that an m-coloured quiver must obey to belong to the mutation class of any D_n quiver. These rules replace the need to run sequences of mutations and directly produce all possible Gabriel quivers of m-cluster-tilted algebras of type D_n. The description recovers the known m=1 classification as a special case and applies uniformly for every n and m. A reader interested in representation theory or cluster algebras would use the rules to enumerate or recognise these algebras without computing mutation orbits.

Core claim

We present an explicit and purely combinatorial characterization of the m-coloured quivers that appear within the m-coloured mutation class of a quiver of type D_n. Consequently, we derive a comprehensive description of the Gabriel quivers associated with m-cluster-tilted algebras of type D_n. The characterization extends the m=1 result of Vatne to arbitrary m.

What carries the argument

The m-coloured mutation class of D_n quivers, defined by explicit rules on arrow counts, directions, and colour assignments between vertices.

If this is right

The Gabriel quiver of any m-cluster-tilted algebra of type D_n must obey the combinatorial conditions.
For m=1 the conditions reduce exactly to the classification of cluster-tilted algebras of type D_n obtained by Vatne.
The same rules classify the underlying quivers for all m-cluster-tilted algebras of type D_n, independent of the choice of initial seed.
Checking the combinatorial conditions suffices to decide membership in the m-coloured mutation class without performing mutations.

Where Pith is reading between the lines

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

The same style of combinatorial rules may exist for other Dynkin types, allowing uniform descriptions across ADE cases.
The conditions provide a direct test that could be implemented to enumerate all such algebras for fixed small values of n and m.
Knowledge of the allowed quivers supplies a concrete starting point for studying the representation theory or derived equivalences of these algebras.

Load-bearing premise

Every m-coloured quiver obtained by mutating a D_n quiver satisfies the stated combinatorial conditions, and every quiver satisfying those conditions arises from some sequence of such mutations.

What would settle it

A single m-coloured quiver that can be reached from a D_n quiver by m-coloured mutations yet violates at least one of the listed combinatorial rules.

Figures

Figures reproduced from arXiv: 2604.22972 by Claudio Qureshi, Viviana Gubitosi.

**Figure 1.** Figure 1: Complete quivers with 3 and 4 vertices respectively. Given two vertices a and b of Q, we will say that Q is an [a, b]-quasi-complete quiver if Q is a quiver in which every pair of distinct vertices is connected by a pair of unique arrows (one in each direction) except for the vertices a and b that they are not connected to each other. a b . a EE b Y . / ! o . } a OO = b a OO view at source ↗

**Figure 2.** Figure 2: [a, b]-quasi-complete quivers with 2, 3 and 4 vertices respectively. Let I = {x1, . . . , xk} ⊆ Q0. We denote by aQb [x1, . . . , xk] (or just by aQb [I]) the [a, b]-quasicomplete subquiver of Q induced by I view at source ↗

**Figure 3.** Figure 3: A 2-coloured quiver in the class Q A 13,2 . We denote by Q A m the union of all Q A n,m for all n. Observe that any connected subquiver of a quiver Q in Q A m is also in Q A m. Here we list some important properties of this class view at source ↗

**Figure 4.** Figure 4: The Dynkin graph Dn Let Q be the following coloured quiver with arrows only of colour 0 and m whose underlying graph is a Dynkin graph of type Dn. n − 1 x (m) 1 (0) / 2 (m) o (0) / 3 (m) o (0) / · · · (m) o (0) / n − 2 (m) o (0) & (0) 8 n (m) f We call the mutation class of type Dn to the set of all quivers mutation equivalent to Q. By a result of Torkildsen [14], we know that this mutation class is finite… view at source ↗

**Figure 5.** Figure 5: The mutation class of the 2-coloured quiver D4 4. The set Q D n,m In this section we define a special class of m-coloured quivers with n vertices which will turning out the coloured mutation class of type Dn. In the sequel, given a subquiver Q′ we denote by A(Q′ ) the set of all arrows belonging to Q′ and we adopt the following convention concerning decorations on the names of vertices: ⊛ means that the ve… view at source ↗

**Figure 6.** Figure 6: Examples of quivers of type I view at source ↗

**Figure 7.** Figure 7: Example of a quiver of type II Observe that the subquivers Qi can be in the set Q A 1,m, i.e., they can have only one vertex. 4.2. Remark. For type I, observe that neither the triangle (axib) nor (ayj b) appear because we do not have an edge a b in Q. As a consequence, the arrows xi → a and xi → b can have the same colour. The same is true for the arrows yi → a and yi → b. However, if r, k ≥ 1, every cycle… view at source ↗

**Figure 8.** Figure 8: Example of 2-coloured quivers in the class Q D 14,2 . 7. The 0-coloured part of a quiver in the mutation class of Dn As in the An-case, the 0-coloured part of a quiver in the mutation class of Dn plays an important role in the study of the m-cluster tilted algebra of type Dn. In fact, every quiver of an m-cluster tilted algebras of type Dn can be obtained by considering the 0-coloured part of the quiver in… view at source ↗

**Figure 9.** Figure 9: The 0-coloured part of the quivers of view at source ↗

**Figure 10.** Figure 10: An example of a 2-cluster tilted algebra simultaneously of type D14 and A14. Considering a quiver that contains a cycle as shown below; we may fix a planar embedding of it to distinguish between clockwise and counterclockwise oriented arrows. However, it should be noted that this convention is unique only up to reflection, as reversing the embedding would swap the roles of the clockwise and counterclockwi… view at source ↗

**Figure 11.** Figure 11: A 4-coloured quiver from Q D 13,4 (left) and the resulting 0-coloured subquiver (right). Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Conflict of interest There are no conflicts of interests or competing interests. Acknowledgements The authors gratefully acknowledge financial support from CSIC (Comisión Sectorial de Inves… view at source ↗

read the original abstract

In this paper, we present an explicit and purely combinatorial characterization of the $m$-coloured quivers that appear within the $m$-coloured mutation class of a quiver of type $\mathbb{D}_n$. The $m$-coloured mutation, as defined by Buan and Thomas in \cite{BT}, generalises the well-known quiver mutation introduced by Fomin and Zelevinsky \cite{FZ}. Consequently, we derive a comprehensive description of the Gabriel quivers associated with $m$-cluster-tilted algebras of type $\mathbb{D}_n$. Notably, our characterization extends a result by Vatne, \cite{Va}, which we recover when $m=1$.

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 presents an explicit and purely combinatorial characterization of the m-coloured quivers that lie in the m-coloured mutation class of any quiver of type D_n. The authors prove that the stated conditions are invariant under m-coloured mutation (in the sense of Buan-Thomas) and that every quiver satisfying the conditions can be reduced to a standard representative via a finite sequence of such mutations. The result recovers Vatne's characterization when m=1 and is applied to give a complete description of the Gabriel quivers of m-cluster-tilted algebras of type D_n.

Significance. If the characterization is complete, the paper supplies a parameter-free combinatorial description of an entire mutation class, generalizing a known m=1 result and directly yielding the quivers of a family of algebras. This strengthens the toolkit for studying higher cluster categories and m-cluster-tilted algebras without iterative mutation computations.

minor comments (2)

In the statement of the main characterization (presumably Theorem 3.1 or equivalent), the conditions on coloured arrows and cycles would be easier to verify if accompanied by a single explicit diagram for small values (e.g., n=4, m=2).
The reduction argument establishing reachability is described as exhibiting a sequence to a standard representative; a brief remark on how the length of this sequence is bounded independently of the starting quiver would clarify the argument's uniformity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary of our manuscript and for recommending minor revision. The report accurately captures the core contribution: an explicit combinatorial characterization of the m-coloured mutation class of D_n quivers that is invariant under Buan-Thomas m-coloured mutation and reduces every such quiver to a standard representative, recovering Vatne's m=1 case and yielding the Gabriel quivers of m-cluster-tilted algebras of type D_n. Since the report lists no specific major comments or requested changes, we have no individual points to address at this stage and await any further details to incorporate into a revised version.

Circularity Check

0 steps flagged

No significant circularity; characterization is independent and combinatorial

full rationale

The paper claims an explicit combinatorial characterization of m-coloured quivers in the mutation class of D_n type, proved by showing the conditions are preserved under m-coloured mutation (defined externally in Buan-Thomas) and that every such quiver reduces to a standard representative. This relies on the external mutation operation and prior results like Vatne's for m=1, without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. The central theorem is self-contained against the given mutation rules and does not reduce to its own output by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established definitions of quiver mutation and m-coloured mutation; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption m-coloured mutation is defined exactly as in Buan and Thomas
The abstract invokes this definition to generalize the Fomin-Zelevinsky mutation.
standard math The mutation class of a D_n quiver is the standard one from cluster algebra theory
Background assumption used to identify the starting object.

pith-pipeline@v0.9.0 · 5422 in / 1348 out tokens · 79904 ms · 2026-05-08T09:03:39.861298+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages

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