Integrable systems on $Fl_n \times Fl_n \times Fl_n //SU(n)$ and $SU(n)$ tensor product multiplicities

TL;DR

This paper constructs a geometric framework using symplectic quotients and moment maps to compute tensor product multiplicities in the representation theory of SU(n), linking lattice point counts to structure constants.

Contribution

It introduces a new geometric approach to tensor product multiplicities via symplectic quotients and moment maps, connecting lattice point formulas to invariance of polarization.

Findings

Lattice point counts correspond to structure constants of SU(n) representations.

The moment map image is a convex polytope capturing tensor product multiplicities.

The approach generalizes Gelfand-Cetlin polytopes and relates to geometric quantization.

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.