Root differences for A _n

Abstract.

Suppose W is a Weyl group with Φ = Φ(W) a root system of W. The set D of root differences is given by D = {α − β : α, β, ∈ Φ}. We define t(Φ) to be the maximum exponent of the torsion subgroup of $\mathbb{Z}\Phi /\mathbb{Z}C,$ for any $C\,\subseteqq\,D.$ In this article we show that if W is of type A _n , then t(Φ) = 2ⁿ.