A cluster algebra approach to presentations of the monoid of uniform block permutations (Q6623516)

This paper establishes a connection between cluster algebras and presentations of the monoid of uniform block permutations, which is a generalization of a Weyl group. The authors consider a quiver \(\Gamma\) and show that any two such quivers are mutation equivalent (Lemma 3.1). They then describe the mutation class \(\mathcal{M}_{\varepsilon_n}\) of a quiver \(Q\) in this class and associate these quivers with potentials to presentations of the monoid \(\mathcal{P}_{n+1}\) of uniform block permutations on the set \(\{1, 2, \dots, n+1\}\).\N\NThe authors recover some classical and known presentations of \(\mathcal{P}_{n+1}\), including FitzGerald's presentation and Everitt and Fountain's presentation, using this cluster algebra approach. This work extends the connection between cluster algebras and presentations of Weyl groups established in earlier works to the setting of reflection monoids, which are a generalization of Weyl groups.\N\NThe introduction provides a thorough overview of the background and motivation for the work. The authors situate their results within the broader context of the connections between cluster algebras and presentations of algebraic structures, particularly Weyl groups. This helps the reader understand the significance and novelty of the current work.\N\NThe technical details of the paper are well explained. The authors carefully define the relevant concepts, such as quivers with frozen vertices, the monoid of uniform block permutations and the mutation class of the quiver \(\Gamma\). The proofs of the key results, such as Lemma 3.1 and the main theorem, are clear and follow logically from the definitions and previous results.\N\NOne strength of the paper is the authors' ability to recover known presentations of the monoid \(\mathcal{P}_{n+1}\) using their cluster algebra approach. This demonstrates the power and versatility of the techniques developed in the paper. It also highlights how this work can shed new light on the structure of this important class of monoids.\N\NThe paper is well organized, with a clear flow from the introduction to the main results. The use of figures helps the reader visualize the key objects of study.