Longest element of a Coxeter group: Difference between revisions

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 (Davis 2007, Section 4.6, pp. 51–53).

• 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).^[1]

• For any ${\displaystyle w\in W,}$ the length satisfies ${\displaystyle \ell (w_{0}w)=\ell (w_{0})-\ell (w).}$^[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 ${\displaystyle A_{n}}$ (${\displaystyle n\geq 2}$), ${\displaystyle D_{n}}$ for n odd, ${\displaystyle E_{6},}$ and ${\displaystyle I_{2}(p)}$ for p odd.^[2]

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• ${\displaystyle A_{n-1}\colon }$ Symmetric group on n symbols, with respect to the generating set of adjacent transpositions: ${\displaystyle (12),(23),(34),\dots ,(n-1,n):}$ the order reversing permutation,

π(i) = n + 1 − i.

of length n(n − 1)/2, which is not central.

• ${\displaystyle BC_{n}\colon }$ The group of signed permutations: the reflection through the origin, which is central.
• ${\displaystyle D_{n}\colon }$
• ${\displaystyle E_{6},E_{7},E_{8}\colon }$
• ${\displaystyle F_{4}\colon }$
• ${\displaystyle H_{3},H_{4}\colon }$
• ${\displaystyle I_{2}(p)\colon }$ The dihedral group, with respect to reflections ${\displaystyle s,t}$ through two lines that form angle ${\displaystyle \pi /p:}$^[3] an alternating word in ${\displaystyle s,t}$ 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.

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

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