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Pré-Publication, Document De Travail Année : 2023

The minimal monomial lfting of cluster algebras I: branching problems

Résumé

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 $\mathcal{A}$ 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 $\mathcal{A}$ 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.
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Dates et versions

hal-04261564 , version 1 (27-10-2023)
hal-04261564 , version 2 (21-02-2024)

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Luca Francone. The minimal monomial lfting of cluster algebras I: branching problems. 2024. ⟨hal-04261564v2⟩
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