Combinatorics for Coxeter groups of typesB _n andD _n

Abstract

In Section 11.5.2 we saw that the groupB _n is isomorphic to the set of allsigned permutations. These are permutations

$$\displaystyle{w:\{ -n,\ldots,-1,0,1,\ldots,n\} \rightarrow \{-n,\ldots,-1,0,1,\ldots,n\},}$$

such that$w(-i) = -w(i)$ for alli. Notice that this forcesw(0) = 0 and the elementw is completely determined byw(1), …, w(n). In one-line notation, we writew = w(1)⋯w(n) with bars to indicate negative numbers. For example, ifw is determined by$w(1) = -3$,w(2) = 4,w(3) = 5,$w(4) = -1$ andw(5) = 2, we write$w =\bar{ 3}45\bar{1}2$.