Newton-Okounkov bodies for categories of modules over quiver Hecke algebras

TL;DR

This paper constructs Newton-Okounkov bodies for quantum coordinate rings using quiver Hecke algebra representation theory, revealing geometric and combinatorial properties linked to cluster theory and Weyl group elements.

Contribution

It introduces a novel geometric framework for quantum coordinate rings via Newton-Okounkov bodies, connecting cluster theory and Weyl group combinatorics.

Findings

Construction of Newton-Okounkov bodies for quantum coordinate rings.

Characterization of properties like rational points, normal fans, and volumes of the bodies.

Establishment of a link between Nakada's hook formula and cluster variables.

Abstract

We show that for a finite-type Lie algebra $\mathfrak{g}$, the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton-Okounkov bodies associated to the quantum coordinate rings $\Aqnw$. When $\mathfrak{g}$ is simply-laced, we use Kang-Kashiwara-Kim-Oh's monoidal categorification to investigate the cluster theory of these bodies. In particular, our construction yields a simplex $\ds$ for every seed $\s$ of $\Aqnw$. We exhibit various properties of these simplices by characterizing their rational points, normal fans, and volumes. As an application, we prove an equality of rational functions relating Nakada's hook formula with the root partitions associated to cluster variables, suggesting further connections between cluster theory and the combinatorics of fully-commutative elements of Weyl groups.