Longest element of a Coxeter group

In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by w₀. See (Humphreys 1992, Section 1.8: Simple transitivity and the longest element, pp. 15–16) and (Davis 2007, Section 4.6, pp. 51–53).

Properties

A Coxeter group has a longest element if and only if it is finite; "only if" is because the size of the group is bounded by the number of words of length less than or equal to the maximum.
The longest element of a Coxeter group is the unique maximal element with respect to the Bruhat order.
The longest element is an involution (has order 2: $w_{0}^{-1}=w_{0}$ ), by uniqueness of maximal length (the inverse of an element has the same length as the element).^[1]
For any $w\in W,$ the length satisfies $\ell (w_{0}w)=\ell (w_{0})-\ell (w).$ ^[1]
A reduced expression for the longest element is not in general unique.
In a reduced expression for the longest element, every simple reflection must occur at least once.^[1]
If the Coxeter group is finite then the length of w₀ is the number of the positive roots.^[1]
The open cell Bw₀B in the Bruhat decomposition of a semisimple algebraic group G is dense in Zariski topology; topologically, it is the top dimensional cell of the decomposition, and represents the fundamental class.
The longest element is the central element –1 except for $A_{n}$ ( $n\geq 2$ ), $D_{n}$ for n odd, $E_{6},$ and $I_{2}(p)$ for p odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram. ^[2]