On homomorphisms and generically $\tau$-regular components for skewed-gentle algebras

TL;DR

This paper provides explicit bases for homomorphisms between indecomposable modules of skewed-gentle algebras, extends combinatorial tools to this setting, and describes certain components of their representation varieties.

Contribution

It introduces explicit homomorphism bases and extends combinatorial formulas to skewed-gentle algebras, including band modules, enhancing understanding of their module categories.

Findings

Explicit basis for homomorphisms between indecomposable modules.

Extension of fringing and kisses concepts to skewed-gentle algebras.

Description of generically τ-regular components in representation varieties.

Abstract

Let $K$ be an algebraically closed field with $\operatorname{char}(K)\neq 2$, and $A$ a skewed-gentle $K$-algebra. In this case, Crawley-Boevey's description of the indecomposable $A$-modules becomes particularly easy. This allows us to provide an explicit basis for the homomorphisms between any two indecomposable representations in terms of the corresponding admissible words in the sense of Qiu and Zhou. Previously (Geiss, 1999), such a basis was only available when no asymmetric band modules were involved. We also extend a relaxed version of fringing and kisses from Br\"ustle et al. (2020) to the setting of skewed-gentle algebras. With this at hand, we obtain convenient formulae for the E-invariant and g-vector for indecomposable $A$-modules, similar to the known expressions for gentle algebras. Note however, that we allow in our context also band-modules. As an application, we describe the indecomposable, generically $\tau$-regular irreducible components of the representation varieties of $A$ as well as the generic values of the E-invariant between them in terms of tagged admissible words.