2025/05/12 by Michaël Bulois, Bulois, Michaël, Roland Denis +3
Computer Science · #Algebraic Geometry (math.AG) #Algorithms and Data Compression #FOS: Mathematics #Matrix Theory and Algorithms #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2505.08812
openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)×⋯× \mathrm GL_d_s(\mathbb C) on \mathbb Cd_1⊗⋯⊗ \mathbb Cd_s and \mathrm GL_d(\mathbb C) on \bigwedger\mathbb Cd, 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 C4⊗\mathbb C4⊗\mathbb C4. 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 C6⊗\mathbb C6⊗\mathbb C6 in 2 hours. Furthermore, a parallel implementation computes the 64,792 (up to \mathfrak S_3) inequalities for \mathbb C7⊗\mathbb C7⊗\mathbb C7 in 188 hours.