Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type

TL;DR

This paper proves Igusa-Todorov's conjecture by showing that certain module sequences in finite type cluster-tilted algebras correspond to c-vectors of maximal green sequences, linking representation theory and cluster combinatorics.

Contribution

It establishes a connection between maximal forward hom-orthogonal sequences and maximal green sequences in finite type cluster-tilted algebras, confirming a conjecture by Igusa and Todorov.

Findings

Modules form a maximal forward hom-orthogonal sequence.

Dimension vectors correspond to c-vectors of maximal green sequences.

Provides a proof of Igusa-Todorov's conjecture.

Abstract

Let $\Lambda$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $\Lambda$ form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\Lambda$. Thus we give a proof of Igusa-Todorov's conjecture.