longest element of a Coxeter group

• 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: $w\_0^\{-1\}\; =\; w\_0$), by uniqueness of maximal length (the inverse of an element has the same length as the element).{{Harv|Humphreys|1992|loc=[https://books.google.com/books?id=ODfjmOeNLMUC&pg=PA16 p. 16]}}
• For any $w\; \backslash in\; W,$ the length satisfies $\backslash ell(w\_0w)\; =\; \backslash ell(w\_0)\; -\; \backslash ell(w).$
• 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 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 the central element −1 except for $A\_n$ ($n\; \backslash geq\; 2$), $D\_n$ 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. {{Harv|Davis|2007|loc=Remark 13.1.8, p. 259}}

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

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{{refbegin}}

• {{citation | title = The Geometry and Topology of Coxeter Groups

|first = Michael W. | last = Davis | year = 2007 | url = http://www.math.osu.edu/~mdavis/davisbook.pdf | isbn = 978-0-691-13138-2 }}

• {{Citation

|title=Reflection groups and Coxeter groups

|isbn=978-0-521-43613-7

|first=James E.

|last=Humphreys

|authorlink=James E. Humphreys

|year=1992

|publisher=Cambridge University Press

}}

{{refend}}

Category:Coxeter groups