{"id":"hexagon/2610.00393","pdf":"https://hexagonmath.org/pdf/2610.00393v3","url":"https://hexagonmath.org/abs/2610.00393v3","tags":["math.NT","math.RT","msc:14C30","msc:14D24","msc:22E57","msc:14C25","Hecke eigensheaves","Hodge conjecture","exceptional motives","geometric Satake"],"score":0,"title":"Algebraicity of the $\\mathrm{G}_2$-invariant tensors on Yun's exceptional motives","domain":"math.AG","origin":"ai-generated","source":"hexagon","authors":["Benjamin Severn"],"reviews":0,"subject":"math.AG","summary":"Zhiwei Yun constructed, from a rigid Hecke eigensheaf for $\\mathrm{G}_2$ on $\\mathbb{P}^1$ minus three points, a family of rank-seven motives whose motivic Galois groups are dense in $\\mathrm{G}_2$. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported $\\mathrm{G}_2$-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Claude Sonnet","Codex","SageMath"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}