The minimal monomial lfting of cluster algebras I: branching problems

Let $\widehat G \subseteq G$ be complex reductive algebraic groups. The branching problem that aims to study $G$-modules as $\widehat G$-modules is encoded by a collection of branching multiplicities parameterised by pairs of dominant weights. The branching algebra $Br(G,\widehat G)$ is a graded algebra whose dimension of homogeneous components are precisely the branching multiplicities. Here, we endow $Br(G, \widehat G)$ with the structure of a graded upper cluster algebra, for some pair of groups. Our result holds if $\widehat G$ is a Levi subgroup of $G$ or in the tensor product case, that is when $\widehat G$ is the diagonal in $G= \widehat G \times \widehat G$, assuming that $G$ is semisimple and simply connected. This sharpens J.Fei's result who got the same statement for $\widehat G=T$ a maximal torus of $G$ and for $G \subseteq G \times G$, assuming $G$ simple, simply laced and simply connected. To prove our result we develop a new geometric and compbinatorial technique called minimal monomial lifting. Let $Y$ be a complex scheme with cluster structure, $T$ be a complex torus and $\mathfrak{X}$ be a suitable partial compactification of $T \times Y$. The minimal monomial lifting produces a canonically graded upper cluster algebra $\overline{\mathcal{A}}$ inside ${\mathcal O}_{\mathfrak{X}}(\mathfrak{X})$ which is, in a precise sense, the best candidate to give a cluster structure on $\mathfrak{X}$ compatible with the one on $Y$. We develop some geometric criteria to prove the equality between $\overline{\mathcal{A}}$ and ${\mathcal O}_{\mathfrak{X}}(\mathfrak{X})$, which doesn't always hold and has some remarkable consequences. This technique is very flexible and will be used elsewhere to endow other classical algebras with the structure of a graded upper cluster algebra.