Diagram calculus for affine Coxeter systems and properties of their associated root systems

Coxeter groups play a central role in algebra, geometry, and combinatorics, as they often underlie many important algebraic structures. In this thesis, we focus on affine Coxeter systems of types $\widetilde{D}$ and $\widetilde{B}$. One of the main objects of investigation is the Temperley--Lieb algebra. In particular, we provide diagrammatic representations of the generalized Temperley--Lieb algebras of types $\widetilde{D}$ and $\widetilde{B}$ by defining infinite-dimensional algebras consisting of decorated diagrams that are isomorphic to the corresponding Temperley--Lieb algebras. To prove the faithfulness of these representations, we make use of special elements in the corresponding Coxeter groups that are irreducible under the so-called star operation. Accordingly, in this work, we present a classification of star irreducible elements for types $\widetilde{D}$ and $\widetilde{B}$. Another aspect explored in the thesis concerns the root systems of affine Coxeter groups. In particular, we extend the reflection representation of a Coxeter system of type $\widetilde{D}$ to the k-th symmetric tensor power of a Euclidean vector space. In order to do so, we define a bijection with a generalization of the diagrammatic representation of the Temperley--Lieb algebra of type $\widetilde{D}$. Finally, we address two problems concerning the rationality of the Poincaré series of reflections in Coxeter groups, motivated by a question proposed by Stembridge in 1997. The first concerns the language of palindromic reduced words, for which we construct an automaton recognizing the language of reflection-prefixes. The second deals with finding explicit formulas for the Poincaré series in the case of affine Coxeter groups.