Polynomial Divided Difference Operators Satisfying the Braid Relations

TL;DR

This paper classifies all polynomial divided difference operators that satisfy braid relations, ensuring well-defined polynomial families indexed by permutations or similar objects.

Contribution

It provides a complete characterization of polynomial divided difference operators satisfying braid relations, clarifying their structure and conditions for well-defined polynomial families.

Findings

Identifies all polynomial divided difference operators satisfying braid relations

Establishes conditions for operators to produce well-defined polynomial families

Contributes to the theoretical understanding of permutation-indexed polynomial families

Abstract

Many interesting families of polynomials are indexed by permutations or related objects, and are defined by applying divided difference operators, modified by polynomials, on some initial base case. The fact that these constructions produce well-defined polynomials is based on the applied modified divided difference operators satisfying the braid relations. We thus determine all the families of polynomial divided difference operators that satisfy the braid relations.