Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.
Decomposition Numbers of Chevalley Groups
