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.
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: ), by uniqueness of maximal length (the inverse of an element has the same length as the element). 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.
See also Coxeter element, a different distinguished element Coxeter number Length function
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