A higher homological approach to the $q$-characters of representations of quantum affine algebras

Abstract

For any acyclic quiver $Q$ without multiple edges, we construct a monoidal category $\mathcal{R}_Q$ whose indecomposable objects are tensor products (over the base field) of finite-dimensional modules over the path algebra of $Q$. We show the existence and uniqueness up to homotopy of certain distinguished chain complexes satisfying good homological properties (higher almost split complexes) preserved under tensoring by objects in $\mathcal{R}_Q$. As a crucial ingredient for this construction, we establish the existence of a family of complete exceptional sequences in $\mathrm{mod}\,\mathbf{k}Q$ satisfying many good properties, which we believe might be of independent interest. We then prove that when $Q$ admits a height function, the Euler characteristics of (the images under certain additive functor of) these complexes coincide with the truncated $q$-characters of the standard modules in Hernandez-Leclerc's category $\mathcal{C}^{(1)}$. Applying our results to the case where the underlying graph of $Q$ is a Dynkin diagram of type $A_n, n \geq 1$, we also interpret the cluster characters of all cluster variables in the finite type cluster algebra $\mathcal{A}_Q$ as Euler characteristics of certain chain complexes in $\mathcal{R}_Q$.