Overview of the Heisenberg--Weyl Algebra and Subsets of Riordan Subgroups

TL;DR

This paper explores the algebraic structure of the Heisenberg--Weyl algebra, its representations, and the related Riordan groups, highlighting properties of Riordan subgroups and applications in combinatorics.

Contribution

It provides a detailed analysis of the connections between the Heisenberg--Weyl algebra and Riordan groups, introducing new properties of Riordan subgroups and their combinatorial applications.

Findings

Characterization of Riordan matrix transformations

Properties of striped Riordan subgroups and quasigroups

Applications to combinatorial structures

Abstract

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e. substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the `striped' Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.