The set of dominance-minimal roots
This article examines the set of so-called dominance-minimal, or elementary roots of a Coxeter system. An earlier paper by \textit{B. Brink} and \textit{R. B. Howlett} [Math. Ann. 296, No. 1, 179-190 (1993; Zbl 0793.20036)] showed that this set is finite, but the bound given in that paper was impractically large. This paper vastly improves the earlier result, giving a method whereby the set of dominance-minimal roots may be explicitly constructed. The paper also lists the sets of dominance-minimal roots for certain special Coxeter systems. Since the finiteness of the set of dominance-minimal roots is linked to the fact that Coxeter groups have an automatic structure, the author expects that her results will be useful to those who wish to develop algorithms for Coxeter groups. The article is structured as follows: Section 1 introduces the main results and the motivation for them, and Section 2 gives some of the necessary background material, including the definitions of two partial orders, precedance and dominance. Readers in danger of confusing the two should read this section carefully, especially the comments at the end of the second-last paragraph on page 374. Section 3 outlines the strategy used to characterise the sets of dominance-minimal roots, which is as follows. The set of dominance-minimal roots \(\mathcal E\) is expressed as the disjoint union of sets \(\mathcal E_S\) derived from subsystems \((W_S,S)\) of the original Coxeter system. The article gives a means of calculating \(\mathcal E_S\) inductively from the sets \(\mathcal E_{S'}\) for which \(S'\) is a subset of \(S\). In Section 4, \(\mathcal E_S\) is shown to be empty for a large class of Coxeter subsystems \((W_S,S)\), and the remaining sections calculate \(\mathcal E_S\) explicitly for a large class of those cases still remaining. This is more than sufficient to calculate \(\mathcal E_S\) for any desired subsystem, and hence to calculate \(\mathcal E\) for any finite-rank Coxeter group. The paper contains many easy-to-read diagrams illustrating the sets of dominance-minimal roots so obtained.
scientific article; zbMATH DE number 1213829