Palindromic richness for languages invariant under more symmetries

TL;DR

This paper explores the concept of $G$-richness in infinite words, generalizing classical palindromic richness to words invariant under complex symmetry groups, and provides characterizations and examples.

Contribution

It introduces the notion of $G$-richness for words invariant under group symmetries, extending classical palindromic concepts to broader symmetry groups and providing new characterizations.

Findings

Multiple equivalent characterizations of $G$-rich words

Examples of $G$-rich words demonstrating the concept

Abstract

For a given finite group $G$ consisting of morphisms and antimorphisms of a free monoid $\mathcal{A}^*$, we study infinite words with language closed under the group $G$. We focus on the notion of $G$-richness which describes words rich in generalized palindromic factors, i.e., in factors $w$ satisfying $\Theta(w) = w$ for some antimorphism $\Theta \in G$. We give several equivalent descriptions which are generalizations of know characterizations of rich words (in the terms of classical palindromes) and show two examples of $G$-rich words.