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Nakayama algebras of small homological dimension and pattern avoiding permutations

Viktória Klász, René Marczinzik, Judith Marquardt

Abstract

We give a combinatorial classification of Nakayama algebras of small homological dimension using the Krattenthaler bijection between Dyck paths and 132-avoiding permutations.

Nakayama algebras of small homological dimension and pattern avoiding permutations

Abstract

We give a combinatorial classification of Nakayama algebras of small homological dimension using the Krattenthaler bijection between Dyck paths and 132-avoiding permutations.
Paper Structure (10 sections, 14 theorems, 9 equations, 13 figures)

This paper contains 10 sections, 14 theorems, 9 equations, 13 figures.

Table of Contents

  1. Introcution
  2. Preliminaries
  3. Nakayama algebras
  4. Dyck paths
  5. Shod Nakayama algebras and Dyck paths
  6. Pattern avoiding permutations and the Krattenthaler bijection
  7. Shod Dyck paths
  8. Classification
  9. Enumeration
  10. Outlook

Key Result

Theorem 1.1

The Krattenthaler bijection restricts to a bijection between shod Dyck paths and 132-avoiding permutations that additionally avoid the patterns 4321 and 4231. This bijection further restricts to a bijection between tilted Dyck paths and 132-avoiding permutations that additionally avoid the pattern 3

Figures (13)

  • Figure 1: Coordinates of the indecomposable modules of the Nakayama algebra with Kupisch series $[3,2,1]$.
  • Figure 2: Projective and injective modules.
  • Figure 3: Projective and injective resolutions. Here $\mathop{\mathrm{pdim}}\nolimits(M)=2$ and $\mathop{\mathrm{idim}}\nolimits(N)=2$.
  • Figure 4: Example of a shod Dyck path.
  • Figure 5: Example of a non-shod Dyck path.
  • ...and 8 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 19 more