Bounding reflection length in an affine Coxeter group
Jon McCammond
and T.Kyle Petersen
DOI: 10.1007/s10801-011-0289-1
Abstract
In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections, and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely, we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on \Bbb R n is bounded above by 2 n, and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.
Pages: 711–719
Keywords: keywords Coxeter group; reflection length
Full Text: PDF
References
1. Bessis, D.: The dual braid monoid. Ann. Sci. Ecole Norm. Super. 36(5), 647-683 (2003)
2. Carter, R.W.: Conjugacy classes in the Weyl group. Compos. Math. 25, 1-59 (1972)
3. Dyer, M.J.: On minimal lengths of expressions of Coxeter group elements as products of reflections.
2. Carter, R.W.: Conjugacy classes in the Weyl group. Compos. Math. 25, 1-59 (1972)
3. Dyer, M.J.: On minimal lengths of expressions of Coxeter group elements as products of reflections.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition