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The Set of Dominance-Minimal Roots

✍ Scribed by Brigitte Brink

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 419 KB
Volume: 206
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1997.7418

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✦ Synopsis

If ␣ and ␤ are positive roots in the root system of a Coxeter group W, we say that ␣ dominates ␤ if w␤ is negative whenever w␣ is negative for w g W. We say that ␣ is elementary or dominance-minimal, if it does not dominate any ␤ / ␣. It Ž . is shown by the author and R. B. Howlett Math. Ann. 296, 1993, 179᎐190 that the set E E of dominance-minimal roots is finite if and only if W has finite rank; this is used to show that W is automatic. To limit the size of the relevant automata, and possibly facilitate other Coxeter group algorithms, we give an explicit description of the set of elementary roots.

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