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: ), by uniqueness of maximal length (the inverse of an element has the same length as the element).[1]
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. [2]
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: ), by uniqueness of maximal length (the inverse of an element has the same length as the element).[1]
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. [2]