Étude de quelques familles de $\lambda$-quiddités et minoration de la taille maximale des $\lambda$-quiddités irréductibles sur un corps fini

$\lambda$-quiddities of size $n$ are $n$-tuples of elements from a fixed set that are solutions to a matrix equation which is fundamental in the study of the combinatorics of the modular group and Coxeter’s friezes. To gain further insight into these objects, we use a notion of irreducibility, which allows restricting the study to a limited number of elements that must be determined for each set. Our goal here is to define several families of $\lambda$-quiddities over finite fields and to study their irreducibility properties, with the specific aim of establishing lower bounds on the maximal size of irreducible elements over ${\mathbb{F}}_q$.