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 and.

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).
 * For any $$w \in W,$$ the length satisfies $$\ell(w_0w) = \ell(w_0) - \ell(w).$$
 * 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.
 * If the Coxeter group is finite then the length of w0 is the number of the positive roots.
 * 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.