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Tensor product stabilization in Kac-Moody algebras

Algebra & Discrete Mathematics

Speaker: Viswanath Sankaran, UC Berkeley
Location: 693 Kerr
Start time: Fri, Oct 3 2003, 2:10PM

When studying representations of finite dimensional simple Lie algebras over C and their infinite dimensional analogues (Kac-Moody algebras), one tries to understand how tensor products of irreducible representations decompose into direct sums of other irreducible representations. For the classical Lie algebra , if denotes the irreducible representation with highest weight , then , where the 's are the well known Littlewood Richardson coefficients. It is a nice fact that the stabilize (i.e become constant) for large .

In this talk, we will consider a larger class of series of Kac-Moody algebras. This class includes , but contains many more series (e.g) , , . I will define the latter Kac-Moody algebras and show that for these, the multiplicities of irreducible representations in tensor product decompositions still exhibit a stabilization behavior. We'll use Littelmann's path model to do this.

The stable values of these multiplicities can be used as structure constants to define a stable tensor product'' operation on a space that could be called the stable representation ring''. Lastly, we'll show that this multiplication operation is indeed associative, making a bonafide algebra that captures tensor products in the limit