An Algorithm to compute the Kronecker cone and other moment cones
Michaël Bulois, Roland Denis, Nicolas Ressayre
TL;DR
<3-5 sentence high-level summary> The paper introduces a novel algorithm to compute the minimal set of linear inequalities defining moment cones for complex reductive-group representations, with a focus on the Kronecker and fermionic cones. It combines a Weyl-group–driven combinatorial framework with a birationality criterion to replace costly convex-geometry computations, enabling computations at unprecedented scale (e.g., Kron(6,6,6) and Kron(7,7,7)). The method consists of five steps: generating a finite tau list, filtering taus, generating Weyl-group elements, selecting dominant maps, and testing birationality via ramification-divisor contraction and compactifications, along with several speedups/filters (BKR, Gröbner, linear-triangular). The authors provide a Python-Sage implementation and demonstrate substantial performance gains, highlighting potential applications to quantum marginal problems and broader geometric-invariant-theory tasks. The work thus advances the computational frontier for moment-cone descriptions and offers practical tools for related algebraic-geometric and representation-theoretic questions.
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.