The Eigencone and Saturation for Spin(8)

Abstract

We explicitly calculate the system of restricted triangle inequalities for the group PSO(8) given by Belkale-Kumar, thereby explicitly solving the eigenvalues of a sum problem for this group (equivalently describing the side-lengths of geodesic triangles in the corresponding symmetric space for the metric d∆ with values in the Weyl chamber ∆). We then apply some computer programs to verify the saturation conjecture for the decomposition of tensor products of finite-dimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights (λ, μ, ν) such that λ + μ + ν is in the root lattice, and any positive integer N , (V (λ)⊗ V (μ)⊗ V (ν)) 6= 0 if and only if (V (Nλ)⊗ V (Nμ)⊗ V (Nν)) 6= 0.