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Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words

Annoy Sengupta, Amit Kuber

Abstract

Generalising a recent work of Dequêne et al. on the connection between perfectly clustering words and band bricks over a particular family of gentle algebras, we characterise band bricks over string algebras whose underlying quiver is acyclic in terms of weakly perfectly clustering pairs of words -- a variant of perfectly clustering words. As a consequence, we characterise band semibricks over all such algebras. Furthermore, the combination of our result and a result of Mousavand and Paquette provides an algorithm to determine whether such a string algebra is brick-infinite.

Characterisation of band bricks over certain string algebras and a variant of perfectly clustering words

Abstract

Generalising a recent work of Dequêne et al. on the connection between perfectly clustering words and band bricks over a particular family of gentle algebras, we characterise band bricks over string algebras whose underlying quiver is acyclic in terms of weakly perfectly clustering pairs of words -- a variant of perfectly clustering words. As a consequence, we characterise band semibricks over all such algebras. Furthermore, the combination of our result and a result of Mousavand and Paquette provides an algorithm to determine whether such a string algebra is brick-infinite.
Paper Structure (9 sections, 23 theorems, 21 equations, 12 figures)

This paper contains 9 sections, 23 theorems, 21 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries of String Algebras
  3. Band Bricks for $\Lambda_N$
  4. Finite Traced Poset
  5. From a string algebra to a finite traced poset
  6. Recovering a string algebra from its finite traced poset
  7. Band bricks over string algebras whose underlying quiver is acyclic
  8. Applications
  9. Comments and future directions

Key Result

Theorem 0

dequêne2023generalization Let $N\geq 2$ and $\mathsf A:=\{2,\cdots,N\}$ be an alphabet linearly ordered with the usual ordering. Then a primitive $\mathsf A$-word $\mathsf w$ is perfectly clustering if and only if the band module $B(\varphi(\mathsf w),1,\lambda)$ for some (equivalently, any) $\lambd

Figures (12)

  • Figure 1: $\Lambda_N$ with $\rho=\{a_1b_2,a_2b_3,\cdots,a_{N-2}b_{N-1},b_1a_2,b_2a_3,\cdots,b_{N-2}a_{N-1}\}$
  • Figure 2: $\Lambda_N$ with $\rho=\{a_1b_2,a_2b_3,\cdots,a_{N-2}b_{N-1},b_1a_2,b_2a_3,\cdots,b_{N-2}a_{N-1}\}$
  • Figure 3: $\Gamma$ with $\rho=\{ac,bd\}$
  • Figure 4: $\Gamma'$ with $\rho=\{b_1a_3,a_3b_1\}$
  • Figure 5: $\Gamma"$ with $\rho=\{c_1ba_3,a_1a_3,c_1c_3\}$
  • ...and 7 more figures

Theorems & Definitions (67)

  • Theorem 0
  • Theorem 0
  • Theorem 0
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 57 more