2026/07/18 by Andrei Gornitskii
Mathematics · #math.RT
We study the branching problem for the pair SLn+m(ℂ)⊃ SLn(ℂ)× SLm(ℂ). We describe the corresponding branching rule in terms of a semigroup Σn,m⊂Λ+×ℤN, where Λ+ is the semigroup of dominant weights of SLn+m, and N is the dimension of maximal unipotent subgroup in SLn+m. Let V(λ) be the irreducible representation of SLn+m with the highest weight λ. For every dominant weight λ∈Λ+ the set of all σ∈Σn,m with dominant weight λ parametrizes the irreducible representations in the restriction V(λ)|SLn× SLm of V(λ) to SLn× SLm. We describe the semigroup Σn,m as the semigroup of integral points in some polyhedral cone and we find the inequalities defining this cone.