Physics – Mathematical Physics
Scientific paper
2000-10-31
J.Phys.A34:11095-11105,2001
Physics
Mathematical Physics
12 pages, LaTeX, version to be published
Scientific paper
10.1088/0305-4470/34/49/324
Information on su(N) tensor product multiplicities is neatly encoded in Berenstein-Zelevinsky triangles. Here we study a generalisation of these triangles by allowing negative as well as non-negative integer entries. For a fixed triple product of weights, these generalised Berenstein-Zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so-called virtual triangles. Inequalities satisfied by the coefficients of the virtual triangles describe a polytope. The tensor product multiplicities may be computed as the number of integer points in this convex polytope. As our main result, we present an explicit formula for this discretised volume as a multiple sum. As an application, we also address the problem of determining when a tensor product multiplicity is non-vanishing. The solution is represented by a set of inequalities in the Dynkin labels. We also allude to the question of when a tensor product multiplicity is greater than a given non-negative integer.
Rasmussen Jorgen
Walton Mark A.
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