Longest element of a Coxeter group

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. ^{[1] [2]}

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: ${\displaystyle w_{0}^{-1}=w_{0}}$), by uniqueness of maximal length (the inverse of an element has the same length as the element). ^[3]
• For any ${\displaystyle w\in W,}$ the length satisfies ${\displaystyle \ell (w_{0}w)=\ell (w_{0})-\ell (w).}$ ^[3]
• 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. ^[3]
• If the Coxeter group is finite then the length of w₀ is the number of the positive roots. ^[3]
• 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 ${\displaystyle A_{n}}$ (${\displaystyle n\geq 2}$), ${\displaystyle D_{n}}$ for n odd, ${\displaystyle E_{6},}$ and ${\displaystyle I_{2}(p)}$ for p odd, when it is −1 multiplied by the order 2 automorphism of the Coxeter diagram. ^[4]

See also

• Coxeter element, a different distinguished element
• Coxeter number
• Length function

References

1. ( Humphreys 1992 , pp.  15–16 , Section 1.8: Simple transitivity and the longest element)
2. ( Davis 2007 , pp. 51–53, Section 4.6)
3. ( Humphreys 1992 , p.  16 )
4. ( Davis 2007 , p. 259, Remark 13.1.8)

• Davis, Michael W. (2007), The Geometry and Topology of Coxeter Groups (PDF), ISBN   978-0-691-13138-2
• Humphreys, James E. (1992), Reflection groups and Coxeter groups (PDF), Cambridge University Press, ISBN   978-0-521-43613-7