Application of graph combinatorics to rational identities of type A

TL;DR

This paper explores how graph combinatorics can be used to compute and understand rational identities of type A, linking algebraic properties of rational functions to graph structures.

Contribution

It introduces a method to compute rational functions associated with words using graph combinatorics and establishes a connection between algebraic factorization and graph disconnecting chains.

Findings

A new combinatorial approach to compute rational functions from words.

A link between numerator factorization and graph disconnecting chains.

Extension of Greene's identities related to the Murnaghan-Nakayama rule.

Abstract

To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).