The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture

Abstract

We introduce the honeycomb model of BZ polytopes, which calculate Littlewood-Richardson coefficients, the tensor product rule for GL(n). Our main result is the existence of a particularly well-behaved honeycomb with given boundary conditions (choice of triple of representations to be tensored together). This honeycomb is necessarily integral, which proves the "saturation conjecture", extending results of Klyachko to give a complete answer to which L-R coefficients are positive. This in turn has as a consequence Horn's conjecture from 1962 characterizing the spectrum of the sum of two Hermitian matrices.

arXiv comment35 pages and 31 pictures - recommended you get the premade PostScript. This version sees mainly expositional changes, plus a conjecture for groups other than GL(n)

References in corpus (2)

• Littlewood-Richardson semigroups
• Equivariant de Rham Theory and Graphs

Cited by in corpus (8)

• FI-modules and stability for representations of symmetric groups
• Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes
• Introduction to twisted commutative algebras
• Symmetric quivers, invariant theory, and saturation theorems for the classical groups
• Littlewood-Richardson coefficients via mirror symmetry for cluster varieties
• The Horn inequalities from a geometric point of view
• Geometry of canonical bases and mirror symmetry
• Generalised Stretched Littlewood-Richardson Coefficients