Coxeter complex

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In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building.

Construction

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The canonical linear representation

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The first ingredient in the construction of the Coxeter complex associated to a Coxeter system   is a certain representation of  , called the canonical representation of  .

Let   be a Coxeter system with Coxeter matrix  . The canonical representation is given by a vector space   with basis of formal symbols  , which is equipped with the symmetric bilinear form  . In particular,  . The action of   on   is then given by  .

This representation has several foundational properties in the theory of Coxeter groups; for instance,   is positive definite if and only if   is finite. It is a faithful representation of  .

Chambers and the Tits cone

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This representation describes   as a reflection group, with the caveat that   might not be positive definite. It becomes important then to distinguish the representation   from its dual  . The vectors   lie in   and have corresponding dual vectors   in   given by

 

where the angled brackets indicate the natural pairing between   and  .

Now   acts on   and the action is given by

 

for   and any  . Then   is a reflection in the hyperplane  . One has the fundamental chamber  ; this has faces the so-called walls,  . The other chambers can be obtained from   by translation: they are the   for  .

The Tits cone is  . This need not be the whole of  . Of major importance is the fact that   is convex. The closure   of   is a fundamental domain for the action of   on  .

The Coxeter complex

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The Coxeter complex   of   with respect to   is  , where   is the multiplicative group of positive reals.

Examples

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Finite dihedral groups

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The dihedral groups   (of order 2n) are Coxeter groups, of corresponding type  . These have the presentation  .

The canonical linear representation of   is the usual reflection representation of the dihedral group, as acting on an  -gon in the plane (so   in this case). For instance, in the case   we get the Coxeter group of type  , acting on an equilateral triangle in the plane. Each reflection   has an associated hyperplane   in the dual vector space (which can be canonically identified with the vector space itself using the bilinear form  , which is an inner product in this case as remarked above); these are the walls. They cut out chambers, as seen below:

 

The Coxeter complex is then the corresponding  -gon, as in the image above. This is a simplicial complex of dimension 1, and it can be colored by cotype.

The infinite dihedral group

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Another motivating example is the infinite dihedral group  . This can be seen as the group of symmetries of the real line that preserves the set of points with integer coordinates; it is generated by the reflections in   and  . This group has the Coxeter presentation  .

In this case, it is no longer possible to identify   with its dual space  , as   is degenerate. It is then better to work solely with  , which is where the hyperplanes are defined. This then gives the following picture:

 

In this case, the Tits cone is not the whole plane, but only the upper half plane. Taking the quotient by the positive reals then yields another copy of the real line, with marked points at the integers. This is the Coxeter complex of the infinite dihedral group.

Alternative construction of the Coxeter complex

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Another description of the Coxeter complex uses standard cosets of the Coxeter group  . A standard coset is a coset of the form  , where   for some proper subset   of  . For instance,   and  .

The Coxeter complex   is then the poset of standard cosets, ordered by reverse inclusion. This has a canonical structure of a simplicial complex, as do all posets that satisfy:

  • Any two elements have a greatest lower bound.
  • The poset of elements less than or equal to any given element is isomorphic to the poset of subsets of   for some integer n.

Properties

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The Coxeter complex associated to   has dimension  . It is homeomorphic to a  -sphere if W is finite and is contractible if W is infinite.

Every apartment of a spherical Tits building is a Coxeter complex.[1]

See also

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References

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Sources

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  • Peter Abramenko and Kenneth S. Brown, Buildings, Theory and Applications. Springer, 2008.