Basics on positively multiplicative graphs and algebras

TL;DR

This paper introduces positively multiplicative graphs and algebras, exploring their fundamental properties and connections to complex topics like fusion rules, boundary descriptions of graded graphs, and random walks on tilings.

Contribution

It presents the basic properties of positively multiplicative graphs and algebras, illustrating their relevance to advanced combinatorial and algebraic problems.

Findings

Positively multiplicative graphs have specific algebraic embedding properties.

Examples demonstrate the relation to fusion rules and boundary problems.

Connections to random walks on alcove tilings are discussed.

Abstract

An oriented graph is said positively multiplicative when its adjacency matrix $A$ embeds in a matrix algebra admitting a basis $\mathsf{B}$ with nonnegative structure constants in which the matrix of the multiplication by $A$ coincides with $A$. The goal of this paper is to present basic properties of this notion and explain, through various simple examples, how it relates to highly non trivial problems like the combinatorial description of fusion rules, the description of the minimal boundary of graded graphs or the study of random walks on alcove tilings.