Automorphism groups of root system matroids

Given a root system R , the vector system R ̃ is obtained by taking a representative v in each antipodal pair { v , − v } . The matroid M ( R ) is formed by all independent subsets of R ̃ . The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root system matroids M ( R ) are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root system matroids.