A categorification of cluster algebras of type B and C through symmetric quivers

In this thesis we present a categorification of cluster algebras of type $B$ and $C$ through a specific class of symmetric quivers arising from triangulations of polygons. Let $\Poly_{2n+2}$ be the regular polygon with $2n+2$ vertices. Let $\theta$ be the rotation of $180^\circ$. Fomin and Zelevinsky showed that $\theta$-invariant triangulations of $\Poly_{2n+2}$ are in bijection with the clusters of cluster algebras of type $B_n$ and $C_n$. Furthermore, cluster variables correspond to the orbits of the action of $\theta$ on the diagonals of the polygon. Given a $\theta$-invariant triangulation $T$ of $\Poly_{2n+2}$, we define cluster algebras of type $B_n$ and $C_n$ with principal coefficients in $T$, and we prove an expansion formula for the cluster variable $x_{ab}$ corresponding to the $\theta$-orbit $[a,b]$ of the diagonal which connects the vertices $a$ and $b$. The formula we present is given in a combinatorial way. On the one hand, it expresses each cluster variable of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$, on the other hand, it allows us to get its expansion in terms of the cluster variables of the initial seed. Moreover, we associate to each $\theta$-orbit $[a,b]$ of $\Poly_{2n+2}$ a modified snake graph $\mathcal{G}_{ab}$, constructed by gluing together the snake graphs corresponding to particular diagonals, obtained from those of $[a,b]$ by identifying some vertices of the polygon. Then we get the cluster expansion of $x_{ab}$ in terms of perfect matchings of $\mathcal{G}_{ab}$. This extends the work of Musiker for cluster algebras of type $B$ and $C$ to every seed. On the other hand, the representation theory of symmetric quivers was developed by Derksen and Weyman, as well as Boos and Cerulli Irelli. A \emph{symmetric quiver} is a quiver $Q$ with an involution of vertices and arrows which reverses the orientation of arrows. A \emph{symmetric representation} is an ordinary representation of $Q$ equipped with some extra data that forces each dual pair of arrows to act anti-adjointly. Symmetric representations are of two types: $orthogonal$ and $symplectic$. They form an additive category which is not abelian. We associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$. Under this correspondence, cluster variables of type $B_n$ (resp. $C_n$) are in bijection with orthogonal (resp. symplectic) indecomposable representations of $Q$. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers, and we present a conjecture for the cyclic case.