Longest element of a Coxeter group

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Not to 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 (Humphreys 1992, Section 1.8: Simple transitivity and the longest element, pp. 15–16) and (Davis 2007, Section 4.6, pp. 51–53).

[edit] 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 $\ell(w_0w) = \ell(w_0) - \ell(w).$ ^[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 a finite Weyl group 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. ^[2]