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

Christof Geiß

TL;DR

The paper develops a comprehensive combinatorial framework for skewed-gentle algebras over fields with $\operatorname{char}(K)\neq 2$, enabling explicit descriptions of indecomposable modules and Hom-spaces through admissible words. By introducing polarized quivers, h-lines, fringing, and kisses, it yields practical formulas for the $E$-invariant and $g$-vectors, and accommodates band-modules. It then applies these tools to classify indecomposable generically $\tau$-regular irreducible components of representation varieties via tagged admissible words, connecting local combinatorics to global geometric structure. Overall, the work provides a detailed, computable bridge from word combinatorics to module theory and geometric components, with implications for cluster-like phenomena in skewed-gentle settings.

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 Brüstle 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 $τ$-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.

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

TL;DR

The paper develops a comprehensive combinatorial framework for skewed-gentle algebras over fields with , enabling explicit descriptions of indecomposable modules and Hom-spaces through admissible words. By introducing polarized quivers, h-lines, fringing, and kisses, it yields practical formulas for the -invariant and -vectors, and accommodates band-modules. It then applies these tools to classify indecomposable generically -regular irreducible components of representation varieties via tagged admissible words, connecting local combinatorics to global geometric structure. Overall, the work provides a detailed, computable bridge from word combinatorics to module theory and geometric components, with implications for cluster-like phenomena in skewed-gentle settings.

Abstract

Let be an algebraically closed field with , and a skewed-gentle -algebra. In this case, Crawley-Boevey's description of the indecomposable -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 Brüstle 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 -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 -regular irreducible components of the representation varieties of as well as the generic values of the E-invariant between them in terms of tagged admissible words.
Paper Structure (40 sections, 18 theorems, 136 equations, 4 tables)

This paper contains 40 sections, 18 theorems, 136 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Context
  3. Overview
  4. Polarized Quivers and their combinatorics
  5. Polarized quivers
  6. Letters
  7. Words
  8. Lexicographic order
  9. Strings and Bands
  10. Projective strings, injective strings, and the operator τ
  11. Example
  12. Admissible words for Q
  13. Orders within strings and bands
  14. Q and construction of A(w) for strings and bands
  15. Admissible words vs. strings and bands
  16. ...and 25 more sections

Key Result

Proposition 3.1

Let ${\underline{Q}}$ be a skewed-gentle polarized quiver and $F\colon{\widehat{{\underline{Q}}}}\rightarrow{\underline{Q}}$ the canonical morphism starting in the corresponding gentle polarized quiver ${\widehat{{\underline{Q}}}}$. Then $\mathbf{x}\in\operatorname{St}({\widehat{{\underline{Q}}}})\c

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Proposition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 4.1
  • Definition 4.2
  • Remark 4.3
  • Lemma 4.4
  • proof
  • ...and 48 more