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

Abstract

We show that for a finite-type Lie algebra $\mathfrak{g}$, Kang-Kashiwara-Kim-Oh's monoidal categorification $\mathcal{C}_w$ of the quantum coordinate ring $\mathcal{A}_{\mathfrak{g}}(\mathfrak{n}(w))$ provides a natural framework for the construction of Newton-Okounkov bodies. In particular, this yields for every seed $\mathcal{S}$ of $\mathcal{A}_{\mathfrak{g}}(\mathfrak{n}(w))$ a simplex $\Delta_{\mathcal{S}}$ of codimension $1$ in $\mathbb{R}^{l(w)}$. We exhibit various geometric and combinatorial properties of these simplices by characterizing their rational points, their normal fans, and their volumes. The key tool is provided by the explicit description in terms of root partitions of the determinantial modules of a certain seed in $\mathcal{S}$ of $\mathcal{A}_{\mathfrak{g}}(\mathfrak{n}(w))$ constructed in \cite{GLS,KKKO}. This is achieved using the recent results of Kashiwara-Kim \cite{KK}. As an application, we prove an equality of rational functions involving root partitions for cluster variables. It implies an expression of the Peterson-Proctor hook formula in terms of heights of monoidal cluster variables in $\mathcal{C}_w$, suggesting further connections between cluster theory and the combinatorics of fully-commutative elements of Weyl groups.