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: ${\displaystyle w_{0}^{-1}=w_{0))$), by uniqueness of maximal length (the inverse of an element has the same length as the element).^[1]
• For any ${\displaystyle w\in W,}$ the length satisfies ${\displaystyle \ell (w_{0}w)=\ell (w_{0})-\ell (w).}$^[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 ${\displaystyle A_{n))$ (${\displaystyle n\geq 2}$), ${\displaystyle D_{n))$ for n odd, ${\displaystyle E_{6},}$ and ${\displaystyle I_{2}(p)}$ for p odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram. ^[2]