ETD

• algebraic geometry
• cluster algebras
• Littlewood-Richardson coefficients
• quiver with potential
• representation theory
• semi-invariant rings

Given a reductive algebraic group G over an algebraically closed field, a reductive subgroup H and an irreducible representation V of G, the branching problem consists in decomposing V under the action of H. In the first part of the thesis, we study this problem in the case of a spherical pair of minimal rank. In particular, we extend by geometric methods a well-known result about the tensor product decomposition. In the second part we study a paper of JiaRui Fei called "Cluster algebras and semi-invariant rings I. Triple flags". In this work the author proves that the semi-invariant ring of quiver representations of the triple flag quiver for the standard dimension vector is an upper cluster algebra associated to the ice hive quiver. In the context of studying the branching problem, this is an important contribution for the study of the Littlewood-Richardson coefficients and maybe a possible interpretation of the hive model due Knutson and Tao.