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A remark on s-torsion pairs and on the lattice of Dyck paths

Baptiste Rognerud

Abstract

There are three classical lattices on the Catalan numbers: the Tamari lattice, the lattice of noncrossing partitions and the lattice of Dyck paths. The first is known to be isomorphic to the lattice of torsion classes of the path algebra of an equioriented quiver of type $A$ and the second is known to be isomorphic to its lattice of wide subcategories. Inspired by the notion of s-torsion classes of Adachi, Enomoto and Tsukamoto, in this short note we interpret the lattice of Dyck paths as a lattice of subcategories.

A remark on s-torsion pairs and on the lattice of Dyck paths

Abstract

There are three classical lattices on the Catalan numbers: the Tamari lattice, the lattice of noncrossing partitions and the lattice of Dyck paths. The first is known to be isomorphic to the lattice of torsion classes of the path algebra of an equioriented quiver of type and the second is known to be isomorphic to its lattice of wide subcategories. Inspired by the notion of s-torsion classes of Adachi, Enomoto and Tsukamoto, in this short note we interpret the lattice of Dyck paths as a lattice of subcategories.

Paper Structure

This paper contains 2 sections, 12 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $\omega$-torsion pairs

Key Result

Proposition 1.1

Let $(P,\leq)$ be a finite poset and $A$ its incidence algebra over a field $\mathbf{k}$. The poset of $\omega$-torsion pairs of $A$ is isomorphic to the distributive lattice of order ideals of $(P,\leq)^{op}$.

Figures (1)

  • Figure 1: First posets of intervals of total orders.

Theorems & Definitions (26)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3: Geyer
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 16 more