Dynkin diagram sequences, stable tensor products,
and representation rings

Viswanath Sankaran

The classical Dynkin diagrams A_n correspond to the Lie algebras sl_n+1C. Highest weights of finite dimensional irreducible representations of sl_n+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 A_n. 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.