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Elements of Order Coxeter Number +1 in Chevalley Groups

  • Bomshik Chang (a1)

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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) matrix

has order 2(h + 1) in SL n+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

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References

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1. Carter, R. W., Simple groups of Lie type (John Wiley, New York, 1972).
2. J.-P., Serre, Cohomologie des groupes discrets Séminaire Bourbaki 399 (Springer Verlag, New York, 1971).
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Elements of Order Coxeter Number +1 in Chevalley Groups

  • Bomshik Chang (a1)

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