Basics on positively multiplicative graphs and algebras
Jérémie Guilhot, Cédric Lecouvey, Pierre Tarrago
TL;DR
This work studies positively multiplicative structures, unifying fusion algebras, adjacency algebras and stochastic processes through PM algebras and PM graphs ($PM$ algebras with a PM-basis). It develops criteria to recognize multiplicativity, introduces the expansion of graphs to infinite graded systems, and links extremal positive harmonic functions on expansions to Perron–Frobenius data within a Kerov–Vershik framework. A substantial portion is devoted to concrete instances, notably Kirillov–Reshetikhin (KR) crystals of affine type A, and to quotients of symmetric-polynomial algebras that yield PM graphs with rich combinatorial structure. The results supply a cohesive toolkit for translating between adjacency-structure positivity and algebraic realizations, with applications to fusion rules, boundary descriptions and random walks on combinatorial lattices.
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.