On embeddability of Coxeter groups into the Riordan group
Tian-Xiao He, Nikolai A. Krylov
TL;DR
This work investigates when finite groups, particularly Coxeter groups, can be realized as subgroups of the Riordan group $\mathcal{R}(\mathbb{K})$. It establishes a strong negative result: for $\mathbb{C}$, any Coxeter group generated by two involutions with a finite product order $m\ge 3$ cannot embed into $\mathcal{R}(\mathbb{C})$, implying non-embeddability for all $\mathcal{S}_n$ and $\mathcal{D}_n$ with $n\ge 3$, while highlighting that embeddings can arise in modular settings. The paper provides concrete positive constructions in $\mathcal{R}(\mathbb{Z}_n)$, showing that dihedral groups $\mathcal{D}_n$ embed and that $\mathcal{S}_3$ has faithful representations over $\mathbb{Z}_3$ and over $\mathrm{GF}(3^q)$, including a complete classification of degree-2 induced representations over $\mathbb{Z}_3$. It also leverages truncations $\mathcal{R}_n(\mathbb{K})$ to relate infinite Riordan arrays to finite matrix groups, and ends with open questions about higher-degree representations and other finite groups. The work thus clarifies how the base ring's characteristic governs embeddability into Riordan groups and provides explicit construction methods for dihedral and small symmetric groups within Riordan arrays.
Abstract
We discuss examples of linear representations of finite groups as subgroups of the Riordan group. In particular, we show that the symmetric group of degree three has no faithful representation as a subgroup of the Riordan group over the complex numbers, but can be embedded as a subgroup of the Riordan group over a field of characteristic three.