**Dynkin diagram sequences, stable tensor products, **

and representation rings

Viswanath Sankaran

The classical Dynkin diagrams *A*_{n} correspond to the Lie algebras
*sl*_{n+1}**C**. Highest weights of finite dimensional irreducible
representations of *sl*_{n+1}**C** 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.