A continuous associahedron of type A

TL;DR

This paper introduces a convex, continuous analogue of the type A associahedron inspired by representation theory, extending discrete cluster structures into a continuous setting with new methods.

Contribution

It constructs a continuous associahedron that retains key properties of the discrete version, including convexity and cluster theory, and embeds classical associahedra within it.

Findings

The continuous associahedron is convex.

Clusters correspond to boundary points.

Embeddings of classical associahedra are established.

Abstract

Taking a representation-theoretic viewpoint, we construct a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. We show that our associahedron shares important properties with the generalized associahedron of type A. Our continuous associahedron is convex and manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. This requires development of several methods that are continuous analogues of discrete methods. We conclude the paper by showing that there is a sequence of embeddings of type A generalized associahedra into our continuous associahedron.