The minimal monomial lfting of cluster algebras I: branching problems
Résumé
Let $Y$ be a complex scheme with cluster structure, $T$ be a complex torus and $\mathfrak{X}$ be a suitable partial compactification of $T \times Y$. We produce, through a technique called minimal monomial lifting, 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. As a special case, through this construction, we recover in a purely geometric and combinatorial method some cluster algebras already appearing in the literature. If $\widehat G$ is a complex reductive subgroup of the complex, semisimple, simply connected algebraic group $G$, we set up a geometric framework to study the branching problem in representation theory, for this pair of groups, through the minimal monomial lifting. We study in detail the case of a Levi subgroup and the tensor product case, for which we have the best possible outcome. Finally, we identify a cluster structure on the base affine space associated to $G$ and discuss some consequences of these results.
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