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
  Bruhat order
  In mathematics, the Bruhat order is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.-History:The Bruhat order on the Schubert varieties of a flag manifold or Grassmannian...
  
  .

• 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).

• For any the length satisfies

• 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 a finite Weyl group
  Weyl group
  In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
  
   then the length of w₀ is the number of the positive roots.

• The open cell Bw₀B in the Bruhat decomposition
  Bruhat decomposition
  In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...
  
   of a semisimple algebraic group
  Semisimple algebraic group
  In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
  
   G is dense in Zariski topology
  Zariski topology
  In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
  
  ; topologically, it is the top dimensional cell of the decomposition, and represents the fundamental class
  Fundamental class
  In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
  
  .

• The longest element is the central element –1 except for (), for n odd, and for p odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram.

See also

• Coxeter element, a different distinguished element
• Coxeter number
  Coxeter number
  In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hence also of a root system or its Weyl group. It is named after H.S.M. Coxeter.-Definitions:...
  
  
• Length function
  Length function
  In mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.-Definition:...