2023/10/18 by Luca Francone, Francone, Luca · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2310.11808
openalex publication_date 2023/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \widehat G ⊆ 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 × \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 ⊆ G × 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 \mathfrakX be a suitable partial compactification of T × Y. The minimal monomial lifting produces a canonically graded upper cluster algebra A inside \mathcal O_\mathfrakX(\mathfrakX) which is, in a precise sense, the best candidate to give a cluster structure on \mathfrakX compatible with the one on Y. We develop some geometric criteria to prove the equality between A and \mathcal O_\mathfrakX(\mathfrakX), 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.