Longest element of a Coxeter group

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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.

Contents 1 Properties 2 Examples 3 See also 4 Notes

[edit] Properties

• The longest element of a Coxeter group is the unique maximal element with respect to the Bruhat order.

• If the Coxeter group is a finite Weyl group 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.

[edit] Examples

• Symmetric group on n symbols: the reverse permutation,

π(i) = n + 1 − i.

of length n(n − 1)/2

• The group of signed permutations: the reflection through the origin.

[edit] See also

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

[edit] Notes