Longest element of a Coxeter group
Appearance
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, Section 1.8: Simple transitivity and the longest element, pp. 15–16) and (Davies 2007, Section 4.6, pp. 51–53) .
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: ), by uniqueness of maximal length (the inverse of an element has the same length as the element).[1]
- For any the length satisfies [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 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.
- The longest element is central except for (), for n odd, and for p odd.[2]
Examples
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- Symmetric group on n symbols, with respect to the generating set of adjacent transpositions: the order reversing permutation,
- π(i) = n + 1 − i.
- of length n(n − 1)/2, which is not central.
- The group of signed permutations: the reflection through the origin, which is central.
- The dihedral group, with respect to reflections through two lines that form angle [3] an alternating word in of maximal length. For p even this is reflection through the origin (which is central), while for p odd this is reflection in the line halfway between the two given lines (the line through the far-most vertex), which is not central.
See also
- Coxeter element, a different distinguished element
- Coxeter number
- Length function
References
- ^ a b c (Humphreys 1992, p. 16)
- ^ (Davis 2007, Remark 13.1.8, p. 259)
- ^ (Davies 2007, p. 346)
- 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, Cambridge University Press, ISBN 978-0-521-43613-7
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