A diagrammatic approach to string polytopes

TL;DR

This paper establishes a precise criterion linking lattice properties of string polytopes to the integrability of highest weight representations for complex classical groups, confirming a conjecture and revealing geometric degenerations.

Contribution

It proves that string polytopes are lattice polytopes if and only if the associated representation integrates to the group, confirming an earlier conjecture.

Findings

String polytopes are lattice polytopes iff the representation integrates to the group.

Every partial flag variety admits a flat degeneration to a Gorenstein Fano toric variety.

The result confirms a conjecture relating lattice properties and representation integrability.

Abstract

We prove that for every complex classical group $G$ the string polytope associated to a special reduced decomposition and any dominant integral weight $\lambda$ will be a lattice polytope if and only if the highest weight representation of the Lie algebra of $G$ with highest weight $\lambda$ integrates to a representation of $G$ itself. This affirms an earlier conjecture and shows that every partial flag variety of a complex classical group admits a flat projective degeneration to a Gorenstein Fano toric variety.