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Saturation and Irredundancy for Spin(8)
Abstract
We explicitly calculate the triangle inequalities for the group PSO(8). Therefore we explicitly solve the eigenvalues of sum problem for this group (equivalently describing the sidelengths of geodesic triangles in the corresponding symmetric space for the Weyl chambervalued metric). We then apply some computer programs to verify two basic questions/conjectures. First, we verify that the above system of inequalities is irredundant. Then, we verify the ``saturation conjecture'' for the decomposition of tensor products of finitedimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights a, b, c such that a+b+c is in the root lattice, and any positive integer N, the tensor product of the irreducible representations V(a) and V(b) contains V(c) if and only if the tensor product of V(Na) and V(Nb) contains V(Nc).
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