2011/08/31 by Xiang Fu
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Artin group #Combinatorics #Coxeter complex #Coxeter group #Dominance (genetics) #Mathematics #Pure mathematics #Rank (graph theory) #Reflection (computer programming) #The Imaginary #math.RT #msc:20F10 #msc:20F55 #msc:20F65 #semigroups and automata theory
paper · pdf · doi:10.2140/pjm.2013.262.339
published as Pacific J. Math., 262 (2013), no. 2, 339--363
arxiv created 2012/10/19 · openalex publication_date 2013/04/16 · arxiv updated 2013/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Brink and Howlett have introduced a partial ordering, called dominance, on the positive roots in the Tits realization of Coxeter groups (Math. Ann. 296 (1993), 179--190). Recently a concept called ∞-height is introduced to each reflection in an arbitrary Coxeter group W (Edgar, Dominance and regularity in Coxeter groups, PhD thesis, 2009). It is known (Dyer, unpublished) that for all W of finite rank, and for each non-negative n, the set of reflections of ∞-height equal to n is finite. However, it is not clear that the concepts of ∞-height and dominance are related. Here we show that the ∞-height of an arbitrary reflection is equal to the number of positive roots strictly dominated by the positive root corresponding to that reflection. We also give applications of dominance to the study of imaginary cones of Coxeter groups.