THE SET OF DOMINANCE-MINIMAL ROOTS

If alpha and beta are positive roots in the root system of a Coxeter g roup W,we say that alpha dominates beta if w beta is negative whenever w alpha is negative for w epsilon W. We say that alpha is elementary or dominance-minimal, if it does not dominate any beta not equal alpha . It is shown by the author and R. B. Howlett (Math. Ann. 296, 1993, 1 79-190) that the set C of dominance-minimal roots is finite if and onl y if W has finite rank; this is used to show that W is automatic. To l imit the size of the relevant automata, and possibly facilitate other Coxeter group algorithms, we give an explicit description of the set o f elementary roots. (C) 1998 Academic Press.