# The set of dominance-minimal roots

## Author

**Brigitte Brink**

## Status

Research Report 94-43

Date: 23 December 1994

## Abstract

If alpha and beta are positive roots in the root system of a Coxeter group W,
we say that alpha dominates beta if w(beta) is negative whenever w(alpha) is
negative for w in W. We say that alpha is elementary or dominance-minimal, if
it does not dominate any root other than itself. It has been shown that the
set of dominance-minimal roots is finite if W has finite rank, and this can
be 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.

## Key phrases

Coxeter group. dominance.

## AMS Subject Classification (1991)

Primary: 20F55

## Content

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Sydney Mathematics and Statistics