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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 $A_n (= sl_{n+1}(\complex))$, if $V_{\gamma}$ denotes the irreducible representation with highest weight $\gamma$, then $V_{\lambda} \otimes V_{\mu} = \oplus_{\nu} c_{\lambda\mu}^{\,\nu}(n) V_{\nu}$, where the $c_{\lambda\mu}^{\,\nu}$'s are the well known Littlewood Richardson coefficients. It is a nice fact that the $c_{\lambda\mu}^{\,\nu}(n)$ stabilize (i.e become constant) for large $n$.

In this talk, we will consider a larger class of series of Kac-Moody algebras. This class includes $A_n$, but contains many more series (e.g) $E_n$, $F_n$, $G_n$. 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 $\mathcal{R}$ that could be called the ``stable representation ring''. Lastly, we'll show that this multiplication operation is indeed associative, making $\mathcal{R}$ a bonafide $\complex-$ algebra that captures tensor products in the limit $n \rightarrow \infty$