Integrable systems on $F{l}_n\times F{l}_n\times F{l}_n//SU(n)$ and $SU(n)$ tensor product multiplicities

Abstract

We construct a densely defined torus action on the symplectic quotient of the product of three complete flag varieties. The closure of the image of the corresponding moment map is a convex polytope. The dimension of the geometric quantization of this space gives the structure constants of the representation ring of $SU(n)$, which we show is given by counting lattice points in the image of the moment map. Such lattice point formulas for the structure constants were given by Berenstein-Zelevinsky; our results show how such formulas arise geometrically as an example of `invariance of polarization', analogous to the description of the Gelfand-Cetlin polytopes by Guillemin-Sternberg. We outline applications to loop groups and to moduli spaces of vector bundles.

Error in the proof of Theorem (page 6, lines 6-7)

References in corpus (1)

• Spectral curves for the triple reduced product of coadjoint orbits for SU(3)