Longest element of a Coxeter group

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, by uniqueness of maximal length.
• For any the length satisfies
• 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 , for n odd, and for p odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram.