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A category of noncrossing partitions

Kiyoshi Igusa

TL;DR

The paper develops an elementary combinatorial model for the cluster-morphism framework in type $A_n$ by using noncrossing partitions and binary forests. It proves that the classifying space $B{\mathcal{N}}{\mathcal{P}}(n)$ is locally $CAT(0)$, hence a $K(\pi,1)$, and identifies its fundamental group as the picture group $G(A_{n-1})$ with a concrete presentation. It embeds this category into the cubical-category framework, demonstrates a faithful representation into unipotent upper-triangular matrices, and situates ${\mathcal{N}}{\mathcal{P}}(n)$ within the cluster-morphism and Hubery–Krause categorical landscapes. Overall, the work provides an accessible, self-contained combinatorial construction of the noncrossing-partition category in type $A_n$ and clarifies its connections to representation theory and cluster theory.

Abstract

In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type $A_n$. This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}``cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type $A_n$.} We prove that the classifying space of this category is locally $CAT(0)$ and thus a $K(π,1)$. We prove a more general statement that classifying spaces of certain ``cubical categories'' are locally $CAT(0)$. The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9].

A category of noncrossing partitions

TL;DR

The paper develops an elementary combinatorial model for the cluster-morphism framework in type by using noncrossing partitions and binary forests. It proves that the classifying space is locally , hence a , and identifies its fundamental group as the picture group with a concrete presentation. It embeds this category into the cubical-category framework, demonstrates a faithful representation into unipotent upper-triangular matrices, and situates within the cluster-morphism and Hubery–Krause categorical landscapes. Overall, the work provides an accessible, self-contained combinatorial construction of the noncrossing-partition category in type and clarifies its connections to representation theory and cluster theory.

Abstract

In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type . This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}``cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type .} We prove that the classifying space of this category is locally and thus a . We prove a more general statement that classifying spaces of certain ``cubical categories'' are locally . The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9].

Paper Structure

This paper contains 21 sections, 39 theorems, 61 equations, 15 figures.

Table of Contents

  1. Noncrossing partitions and binary forests
  2. Noncrossing partitions
  3. Binary forests
  4. Binary forests as morphisms
  5. Composition of cluster morphisms
  6. Compatibility of edge sets
  7. Binary trees as maximal compatible sets
  8. Proof of Theorem \ref{['thm1: morphisms are maximal compatible sets of edge vectors']}
  9. First case of Theorem \ref{['thm2: bijection ET to E']}
  10. Proof of Theorem \ref{['thm2: bijection ET to E']}
  11. Cube complexes
  12. Cubical categories
  13. Representation of ${\mathcal{N}}{\mathcal{P}}(n)$
  14. ${\mathcal{N}}{\mathcal{P}}(n)$ is cubical
  15. Fundamental group
  16. ...and 6 more sections

Key Result

Theorem 1

A simply connected cubical space is $CAT(0)$ if and only if the link of every vertex is a flag complex.

Figures (15)

  • Figure 1: On the left, two parts cross: $A=\{0,4\}$ and $B=\{2,6\}$ cross since $0<2<4<6$ (equivalently, the closed intervals $[0,4],[2,6]$ overlap). If we take one of these, say $A$, on the right, we cannot merge $1,2$ or $3$ with $5$ or $6$. We can merge several pair on the right as long as these pairs are not crossing. For example we cannot merge $2$ with $A$ and $1$ with $3$. This pair of operations "cross".
  • Figure 2: Example of noncrossing partition of $V=\{1,2,3,4,5,6,7,8\}$ into five parts $A=\{1,6,8\},B=\{2,4\},C=\{3\}, D=\{5\}, E=\{7\}$.
  • Figure 3: Binary tree on 10 vertices. The root is $v_8$. Removing the root gives two subtrees.
  • Figure 4: Two refinements of ${\mathcal{Q}}=\{P,U\}$ given by ${\mathcal{R}}=\{B,W,F,A\}$ where $P=B\cup W$ and $U=A\cup F$ and ${\mathcal{S}}$ which is a further refinement of ${\mathcal{R}}$ given by subdividing $W$ into $W=C\cup D\cup E$.
  • Figure 5: $T\cup \mu_T(S)= C-E,C-D,C-B,A-F$ in binary tree notation (Definition \ref{['def: binary tree notation']}). We can immediately see that these edges are compatible since they can be completed to a binary tree with edges indicated with dotted lines. (See Definition \ref{['def: compatibility for G(V)']} and Theorem \ref{['thm: maximal compatible sets are rooted binary trees']}.)
  • ...and 10 more figures

Theorems & Definitions (94)

  • Theorem 1: Gromov
  • Corollary 2
  • Proposition 1.1
  • proof
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Definition 1.6
  • ...and 84 more