Longest 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 ''w''₀. 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: $w_0^ = w_0$), 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 ''w''₀ is the number of the positive roots.
* The open cell ''Bw''₀''B'' 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

References

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Coxeter groups