Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Selected facts on products of two involutions in the Riordan group

Roksana Słowik, Tejbir Lohan

TL;DR

This work analyzes reversibility in the Riordan group $ R$, showing that not all reversible elements are strongly reversible and that products of two involutions are always commutators (up to sign). It develops a commutator-centric view, proving that in characteristic $0$ the Riordan commutator subgroup consists of arrays with $1$'s on the main diagonal and that every such element is a single commutator. The authors then examine several natural subgroups and demonstrate that, within these subgroups, products of two involutions likewise yield commutators, with explicit involution structures described. They further connect strong reversibility to pseudo-involutions and classify reversible behavior via Ofar’s results on $f(t)$, illustrating the nuanced distinction between reversible elements and products of two involutions. Overall, the paper clarifies the internal algebraic structure of the Riordan group and its subgroups, with implications for generating sets and the interplay between combinatorial representations and algebraic properties.

Abstract

An element of a group is called \emph{reversible} if it is conjugate to its inverse, and \emph{strongly reversible} if it can be expressed as a product of two involutions. We study strongly reversible elements in the Riordan group and in several of its important subgroups. We show that not every reversible element in the Riordan group is strongly reversible, and we investigate products of reversible elements in the Riordan group.

Selected facts on products of two involutions in the Riordan group

TL;DR

This work analyzes reversibility in the Riordan group , showing that not all reversible elements are strongly reversible and that products of two involutions are always commutators (up to sign). It develops a commutator-centric view, proving that in characteristic the Riordan commutator subgroup consists of arrays with 's on the main diagonal and that every such element is a single commutator. The authors then examine several natural subgroups and demonstrate that, within these subgroups, products of two involutions likewise yield commutators, with explicit involution structures described. They further connect strong reversibility to pseudo-involutions and classify reversible behavior via Ofar’s results on , illustrating the nuanced distinction between reversible elements and products of two involutions. Overall, the paper clarifies the internal algebraic structure of the Riordan group and its subgroups, with implications for generating sets and the interplay between combinatorial representations and algebraic properties.

Abstract

An element of a group is called \emph{reversible} if it is conjugate to its inverse, and \emph{strongly reversible} if it can be expressed as a product of two involutions. We study strongly reversible elements in the Riordan group and in several of its important subgroups. We show that not every reversible element in the Riordan group is strongly reversible, and we investigate products of reversible elements in the Riordan group.
Paper Structure (7 sections, 6 theorems, 8 equations, 1 table)

This paper contains 7 sections, 6 theorems, 8 equations, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Products of two involutions
  4. Products of two involutions in subgroups of $\mathcal{R}$
  5. Remarks on reversible elements
  6. Strongly reversible elements
  7. Products of two reversible elements

Key Result

Proposition 2.1

If $F$ is a field of characteristic $0$, then the commutator subgroup of the Riordan group $\mathcal{R}$ defined over $F$ consists of all Riordan arrays with $1$'s in the main diagonal. Moreover, every element of the commutator subgroup is a (single) commutator.

Theorems & Definitions (8)

  • proof : Proof of Fact \ref{['prop:comm']}
  • Proposition 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6