Longest element of a Coxeter group

Short description: Unique element of maximal length in a finite 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) and (Davis 2007).

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).
• For any $w\in W,$ the length satisfies $\ell (w_{0}w)=\ell (w_0)-\ell (w).$
• 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 $A_n$ ($n\ge 2$), $D_n$ 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

References

• Davis, Michael W. (2007), The Geometry and Topology of Coxeter Groups, ISBN 978-0-691-13138-2, http://www.math.osu.edu/~mdavis/davisbook.pdf 
• Humphreys, James E. (1992), Reflection groups and Coxeter groups, Cambridge University Press, ISBN 978-0-521-43613-7