Essential paths space on ADE SU(3) graphs: A geometric approach

TL;DR

This paper provides a geometric framework for understanding path operators on ADE SU(3) graphs, revealing how path spaces decompose into orthogonal subspaces generated by essential paths and triangular cell structures.

Contribution

It introduces a geometric interpretation of path creation and annihilation operators for SU(3) graphs, connecting them to triangular cell structures and path space decomposition.

Findings

Path space decomposes into orthogonal subspaces via creation operators.

Path operators are related to triangular cell structures.

Decomposition applies to paths of any length.

Abstract

For simply laced $SU(3)$ graphs we offer a geometric understanding of the path creation and annihilation operators for $SU(3)$ in terms of creation and annihilation of sequences of three vertices forming triangular cells or collapsed triangular cells. We prove that the space of paths of a given length can be decomposed as a direct sum of orthogonal sub-spaces constructed by recurrent applications of the path creation operator on subspaces of essential paths of shorter length.