Dynkin diagram sequences, stable tensor products,
and representation rings

Viswanath Sankaran

The classical Dynkin diagrams An correspond to the Lie algebras sln+1C. Highest weights of finite dimensional irreducible representations of sln+1C can be indexed by partitions . There are combinatorial formulas for most representation theoretic entities; notably the Littlewood-Richardson rule computes the multiplicity of a given representation in a tensor product. This rule can be used to study how the tensor product decomposition changes with n and shows that the multiplicity of a given representation stabilizes as n grows large.

One can also index highest weights using pairs of partitions, letting our weights be supported on both ends of the diagram An. It turns out that multiplicities still stabilize in this setting. We will show that these results hold for a broad class of sequences of Dynkin diagrams of the form
X-o-o-···-o-Y, though the associated Kac-Moody algebras are usually infinite dimensional and non-affine. The main tool used is Littelmann's path model.