Longest element of a Coxeter group explained

Longest element of a Coxeter group should not be confused with Coxeter 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 ^{-1} _{0} = 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.

See also

• Coxeter element, a different distinguished element
• Coxeter number
• Length function