Structure of the tensor product semigroup (Q863780)

Let \(G\) be a complex reductive Lie group. Then finite-dimensional irreducible representations of \(G\) are parameterized by their highest weights, \(\lambda\mapsto V_{\lambda}\). The authors consider the set \[ Tens(G)=\{(\lambda,\mu,\nu)\mid (V_{\lambda}\otimes V_{\mu}\otimes V_{\nu})^G \neq 0\}. \] This set forms a semigroup with respect to addition. In the paper under review the authors prove several structural results about \(Tens(G)\) and compute \(Tens(Sp(4,\mathbb{C}))\) and \(Tens(G_2)\). One of their main results is the following Theorem: For each complex reductive Lie group \(G\) the set \(Tens(G)\) is a finite union of elementary subsets of \(L^3\), where \(L\) is the character lattice of a maximal (split) torus in \(G\).