An estimate of canonical dimension of groups based on Schubert calculus
Rostislav Devyatov
TL;DR
The paper links the canonical 0-dimension $\mathfrak{cd}(G)$ of split simply connected semisimple groups to the cohomology of full flag varieties via Schubert calculus. It develops Galois-descent machinery and Picard-group behavior to relate torsors and their quotients by Borel subgroups, enabling a multiplicity-free criterion on Schubert divisors to bound $\mathfrak{cd}(G)$. A key theorem shows that if a product of Schubert divisors is multiplicity-free, then $\mathfrak{cd}(G) \le \dim(G/B) - (n_1+\cdots+n_r)$, leading to explicit upper bounds for exceptional types: $\mathfrak{cd}(G) \le 17$ for $E_6$, $\le 37$ for $E_7$, and $\le 86$ for $E_8$. This work provides a novel bridge between torsor rational points, flag-variety geometry, and Schubert-calculus, with concrete consequences for the canonical dimension of exceptional groups.
Abstract
We sketch the proof of a connection between the canonical (0-)dimension of semisimple split simply connected groups and cohomology of their full flag varieties. Using this connection, we get a new estimate of the canonical (0-)dimension of simply connected split exceptional groups of type $E$ understood as a group.