The canonical dimension of a semisimple group and the unimodular degree of a root system

TL;DR

This paper introduces an elementary algorithm to estimate the canonical dimension of split semisimple algebraic groups, confirming known bounds and establishing new ones for various types.

Contribution

It provides a simple, effective algorithm for computing upper bounds on the canonical dimension of split semisimple groups, advancing understanding in algebraic group theory.

Findings

Confirmed existing bounds for certain groups

Produced new bounds for types F4 and E6

Algorithm is short and elementary

Abstract

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko and Devyatov as well as we produce new bounds (e.g. for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).