An Algorithm to compute the Kronecker cone and other moment cones

TL;DR

This paper introduces a new algorithm and implementation for computing the inequalities defining moment cones of complex reductive group representations, enabling calculations at larger scales than previously possible.

Contribution

The authors develop a novel algorithm that overcomes previous limitations, allowing efficient computation of moment cone inequalities for complex representations, including the Kronecker and fermionic cones.

Findings

Computed 5,333 inequalities for the Kronecker cone in 2 hours.

Generated 64,792 inequalities for larger cases in 188 hours.

Overcame previous computational bottlenecks in algebraic geometry and combinatorics.

Abstract

We describe a new algorithm that computes the minimal list of inequalities for the moment cone of any representation of a complex reductive group, with implementation details for two fundamental cases: the Kronecker cone (governing the asymptotic support of Kronecker coefficients) and the fermionic cone. These correspond to the actions of ${\mathrm GL}\_{d\_1}({\mathbb C})\times\cdots\times {\mathrm GL}\_{d\_s}({\mathbb C})$ on ${\mathbb C}^{d\_1}\otimes\cdots\otimes {\mathbb C}^{d\_s}$ and ${\mathrm GL}\_d({\mathbb C})$ on $\bigwedge^r{\mathbb C}^d$, respectively. An implementation for these two cases in Python-Sage is available at https://ea-icj.github.io/. Our work overcomes the fundamental limitations that previously restricted such computations to cases like ${\mathbb C}^4\otimes{\mathbb C}^4\otimes{\mathbb C}^4$. The state-of-the-art method by Vergne-Walter faced two major bottlenecks: one from combinatorial geometry in finite-dimensional vector spaces, and another from deciding whether certain dominant morphisms are birational - a problem in effective algebraic geometry that lacked a direct algorithmic solution. We surmount these obstacles by: a novel use of Weyl group actions to master combinatorial complexity, and an original algorithm for deciding birationality that replaces previous workarounds relying on convex geometry. Our approach allow us to tackle problems at a new scale. We compute the minimal list of 5,333 (up to $\mathfrak S\_3$) inequalities for the Kronecker cone ${\mathbb C}^6\otimes{\mathbb C}^6\otimes{\mathbb C}^6$ in 2 hours. Furthermore, a parallel implementation computes the 64,792 (up to $\mathfrak S\_3$) inequalities for ${\mathbb C}^7\otimes{\mathbb C}^7\otimes{\mathbb C}^7$ in 188 hours.