On embeddability of Coxeter groups into the Riordan group

TL;DR

This paper explores the conditions under which Coxeter groups can be embedded into the Riordan group, providing specific examples and limitations related to symmetric groups and field characteristics.

Contribution

It demonstrates that certain finite groups, like the symmetric group of degree three, cannot be faithfully embedded into the Riordan group over complex numbers but can over fields of characteristic three.

Findings

Symmetric group of degree three cannot be embedded over complex numbers.

Embedding is possible over fields of characteristic three.

Provides examples of linear representations within the Riordan group.

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.