Longest element of a Coxeter group
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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: [math]\displaystyle{ w_0^{-1} = w_0 }[/math]), by uniqueness of maximal length (the inverse of an element has the same length as the element).[1]
- For any [math]\displaystyle{ w \in W, }[/math] the length satisfies [math]\displaystyle{ \ell(w_0w) = \ell(w_0) - \ell(w). }[/math][1]
- 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.[1]
- If the Coxeter group is finite then the length of w0 is the number of the positive roots.[1]
- 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 [math]\displaystyle{ A_n }[/math] ([math]\displaystyle{ n \geq 2 }[/math]), [math]\displaystyle{ D_n }[/math] for n odd, [math]\displaystyle{ E_6, }[/math] and [math]\displaystyle{ I_2(p) }[/math] for p odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram. [2]
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
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