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\newcommand{\Ea}{ {\mathcal E}_n^{\alpha}(q)} \newcommand{\Eak}{ {\mathcal E}_n^{\alpha, \mathcal{K}}(q)} \newcommand{\EE}{ {\mathcal E}_n} \newcommand{\Elambda}{E_{\blambda}} \newcommand{\FF}{ {\mathcal F}_n} \newcommand\bS{\Sigma} \newcommand\bn{\mathbf{n}} \newcommand\blambda{{\boldsymbol\lambda}} \newcommand\btau{{\boldsymbol\tau}} \newcommand\bnu{{\boldsymbol\nu}} \newcommand\be{\mathbb{E}} \newcommand\bmu{{\boldsymbol\mu}} \newcommand{\pco}{\mathcal{SP}^{{\rm ord }}} \newcommand{\co}{{\rm comp}_r(n)} \newcommand{\secu}{{\rm seq}} \newcommand\YYlambda{\mathcal{Y}_{{ \alpha }}(q)} \newcommand\YYalpha{\mathcal{Y}_{{ \alpha} }(q)} \newcommand\beLambda{\be_{\Lambda}} \newcommand{\HRule}{\rule{\linewidth}{0.2mm}} \newtheorem{teo}{ Theorem}[chapter] \newtheorem{coro}{ Corollary}[chapter] \newtheorem{propos}{ Proposition}[chapter] \theoremstyle{plain} \newtheorem{criterion}[teo]{Criterio} \newtheorem{defi}{Definition}[chapter] \newtheorem{exa}{Example} \newtheorem{lem}{Lemma} \newtheorem{obs}{Remark} \newtheorem{axiom}{Axioma} \newenvironment{demo} {\textsc{Proof.}} {\quad \hfill $\Box$} \newenvironment{demoemb} {\textsc{Proof of Theorem \ref{embeddding}.}} {\quad \hfill $\Box$} \newenvironment{problem} \setcounter{tocdepth}{2} \begin{document} \begin{titlepage} \begin{figure}[h!] \centering \includegraphics[width=3.8cm,height=3.5cm]{logo} \end{figure} \vspace*{8mm} \begin{center} {\Large \textbf{Cell structure for the Yokonuma-Hecke algebra and related algebras}}\\[1cm] Jorge Espinoza Espinoza \vspace*{2.5cm} A thesis submitted in partial fulfillment of the requirements\\ for the degree of Doctor of Mathematics \vspace*{6cm} Institute of Mathematics\\ University of Talca \vspace*{1cm} January 2018 \end{center} \end{titlepage} \tableofcontents \notocchapter{Acknowledgements} {\it Sin duda, debo agradecer a muchas personas por hacer posible este trabajo, ya sea bla bla \chapter*{Introduction} \chapter{Preliminaries} \section{The symmetric group} \section{Young tableaux and set partitions} \subsection{Combinatorics of Young tableaux} \subsection{Set partitions} \section{Cellular algebras} \chapter{Representation theory of the Yokonuma-Hecke algebra} \section{Yokonuma Hecke algebra} \section{Tensorial representation of $\Y(q)$} \begin{defi}{\label{tensoraction}} Let $V$ be the free $R$-module with basis $\{v_{i}^{t}\mid 1\leq i\leq n,\; 0\leq t\leq r-1\}$. Then we define operators $\mathbf{T}\in \End_R(V)$ and $\mathbf{G}\in \End_R(V^{\otimes 2})$ as follows: \begin{equation}{\label{operatorT}} (v_i^t)\mathbf{T}:=\xi^{t}v_i^t \end{equation} and \begin{equation}{\label{operatorG}} (v_i^t\otimes v_j^s)\mathbf{G}:=\left\{\begin{array}{ll}v_j^s\otimes v_i^t&\mbox{ if }\;t\neq s\\ qv_i^t\otimes v_j^s&\mbox{ if }\;t=s,\;i=j\\ v_j^s\otimes v_i^t&\mbox{ if }\;t=s,\;i>j\\ (q-q^{-1})v_i^t\otimes v_j^s+v_j^s\otimes v_i^t&\mbox{ if }\;t=s,\;i