Total positivity, cluster algebras and categorification

Let $G$ be the group of unipotent uppertriangular $n \times n$ matrices. For a matrix of this group, being totally positive means to have all its non-trivial minors positive (those which are non-vanishing in $\mathbb{C}[G]$). From an algorithmic point of view, it is interesting to find a subset of the set of the minors, seen as elements of $\mathbb{C}[G]$, which fully characterizes the total positivity. Such subsets, with only $\left( \begin{matrix} n \\ 2 \end{matrix} \right)$ minors, exist. For example, if $$M = \left( \begin{matrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} \right)$$ then, the positivity of the minors $$\left| \begin{matrix} x & y \\ 1 & z \end{matrix} \right| \quad \text{and} \quad x \quad \text{and} \quad y $$ is enough to obtain the positivity of all minors. In the first part of this talk, we will see how to pass from such a positivity criterium to others and to create a familly of such criteria. This processus is encoded by a "cluster algebra". After that, we will see how to attach certain categories to these cluster algebras, which permits to prove, through the representation theory, results which are inaccessible by the combinatoric.