# su(N) tensor-product multiplicities as polytope volumes

*by Jorgen Rasmussen*

*Institution:*University of Melbourne

*Date: Thu 20th October 2005*

*Time: 2:15 PM*

*Location: Theatre 1, Old Geology Building, The University of Melbourne*

*Abstract*: Information on su(N) tensor-product multiplicities is neatly

encoded in the so-called Berenstein-Zelevinsky (BZ) triangles.

The multiplicities for given highest-weight modules are given simply

by the number of allowed BZ triangles associated to these modules.

This illustrates the vital role combinatorics plays in the theory

of highest-weight modules of Lie algebras.

While this talk concerns an explicit formula for these multiplicities,

it only assumes very basic knowledge on the problem.

By relaxing one of the conditions in the definition of the BZ triangles,

one may introduce virtual BZ triangles, enabling

one to characterize a tensor-product multiplicity as the number of integer

points in a convex polytope. It turns out that this discretized volume

can be evaluated explicitly as a multiple sum.

*For More Information:* Iwan Jensen tel. 03 8344-5214 email: i.jensen@ms.unimelb.edu.au