COXETER GROUP
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(Redirected from Affine Weyl group)
In mathematics, a 'Coxeter group', named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections.
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.
Formally, a 'Coxeter group' can be defined as a group with the presentation
:
where ''m''''ii'' = 1 and ''m''''ij'' ≥ 2 for ''i'' ≠ ''j''.
The condition ''m''''ij'' = ∞ means no relation of the form (''r''''i'' ''r''''j'')''m'' should be imposed.
A number of conclusions can be drawn immediately from the above definition.
★ The relation ''m''''ii'' = 1 means that (''r''''i'')2 = 1 for all ''i'' ; the generators are involutions.
★ If ''m''''ij'' = 2, then the generators ''r''''i'' and ''r''''j'' commute. This follows by observing that
::''xx'' = ''yy'' = 1,
: together with
:: ''xyxy'' = 1
: implies that
:: ''xy'' = ''xxyxyy'' = ''yx''.
★ In order to avoid redundancy among the relations, it is necessary to assume that ''mij''=''mji''. This follows by observing that
::''yy'' = 1,
: together with
:: (''xy'')''m'' = 1
: implies that
:: (''yx'')''m'' = (''yx'')''m''''yy'' = ''y''(''xy'')''m''''y'' = ''yy'' = 1.
The 'Coxeter matrix' is the ''n''×''n'', symmetric matrix with entries ''mij''. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal serves to define a Coxeter group.
The Coxeter matrix can be conveniently encoded by a 'Coxeter graph', as per the following rules.
★ The vertices of the graph are labelled by generator subscripts.
★ Vertices ''i'' and ''j'' are connected if and only if ''m''''ij'' ≥ 3.
★ An edge is labelled with the value of ''m''''ij'' whenever it is 4 or greater.
In particular, two generators commute if and only if they are not connected by an edge.
Furthermore, if a Coxeter graph has two or more connected components, the associated group is the direct product of the groups associated to the individual components.
The graph in which vertices 1 through ''n'' are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group ''S''''n''+1; the generators correspond to the transpositions (1 2), (2 3), ... (''n'' ''n''+1). Two non-consecutive transpositions always commute, while (''k'' ''k''+1) (''k''+1 ''k''+2) gives the 3-cycle (''k'' ''k''+1 ''k''+2). Of course this only shows that ''Sn+1'' is a quotient group of the Coxeter group, but it is not too difficult to check that equality holds.
Every Weyl group can be realized as a Coxeter group. The Coxeter graph can be obtained from the Dynkin diagram by replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. The example given above corresponds to the Weyl group of the root system of type ''An''. The Weyl groups include most of the finite Coxeter groups, but there are additional examples as well. The following list gives all connected Coxeter graphs giving rise to finite groups:
Comparing this with the list of simple root systems, we see that ''Bn'' and ''Cn'' give rise to the same Coxeter group. Also, ''G''2 appears to be missing, but it is present under the name ''I''2(6). The additions to the list are ''H''3, ''H''4, and the ''I''2(''p'').
Some properties of the finite Coxeter groups are given in the following table:
All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series ''I''2(''p''). The symmetry group of a regular ''n''-simplex is the symmetric group ''S''''n''+1, also known as the Coxeter group of type ''An''. The symmetry group of the ''n''-cube is the same as that of the ''n''-cross-polytope, namely ''BCn''. The symmetry group of the regular dodecahedron and the regular icosahedron is ''H''3. In dimension 4, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group ''F''4, while the other two have symmetry group ''H''4.
The Coxeter groups of type ''D''''n'', ''E''6, ''E''7, and ''E''8 are the symmetry groups of certain semiregular polytopes.
The affine Weyl groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotient group is itself a Weyl group, and the Coxeter graph is obtained from the Coxeter graph of the Weyl group by adding an additional vertex and one or two additional edges. For example, for ''n'' ≥ 2, the graph consisting of ''n''+1 vertices in a circle is obtained from ''An'' in this way, and the corresponding Coxeter group is the affine Weyl group of ''An''. For ''n'' = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
A list of the Affine Coxeter groups follows:
Note the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.
There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic geometry.
A choice of reflection generators gives rise to a length function ''l'' on a Coxeter group, namely the minimum number of uses of generators required to express a group element. An expression for ''v'' using ''l(v)'' generators is a ''reduced word''. For example, the permutation (13) in ''S3'' has two reduced words, (12)(23)(12) and (23)(12)(23).
Using reduced words one may define two partial orders on the Coxeter group, the ''weak Bruhat order'' and the ''strong Bruhat order''. An element ''v'' exceeds an element ''u'' in the strong Bruhat order if some (or equivalently, any) reduced word for ''v'' contains a reduced word for ''u'' as a substring, where some letters (in any position) are dropped. Whereas ''v ≥ u'' in the weak Bruhat order if some reduced word for ''v'' contains a reduced word for ''u'' as an initial segment.
For example, the permutation (1 2 3) in ''S3'' has only one reduced word, (12)(23),
so covers (12), (23) in the strong order but only covers (12) in the weak order.
★ Larry C Grove and Clark T. Benson, ''Finite Reflection Groups'', Graduate texts in mathematics, vol. 99, Springer, (1985)
★ James E. Humphreys, ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990)
★ Richard Kane, ''Reflection Groups and Invariant Theory'', CMS Books in Mathematics, Springer (2001)
★ Anders Björner and Francesco Brenti, ''Combinatorics of Coxeter Groups'', Graduate texts in mathematics, vol. 231, Springer, (2005)
★ Artin group
★ Weyl group
★ Triangle group
★ Coxeter number
★ Complex reflection group
★ Kazhdan-Lusztig polynomial
★
★ Jenn, software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators.
In mathematics, a 'Coxeter group', named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections.
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.
| Contents |
| Definition |
| An example |
| Finite Coxeter groups |
| Symmetry groups of regular polytopes |
| Affine Weyl groups |
| Hyperbolic Coxeter groups |
| Bruhat order |
| References |
| See also |
| External links |
Definition
Formally, a 'Coxeter group' can be defined as a group with the presentation
:
where ''m''''ii'' = 1 and ''m''''ij'' ≥ 2 for ''i'' ≠ ''j''.
The condition ''m''''ij'' = ∞ means no relation of the form (''r''''i'' ''r''''j'')''m'' should be imposed.
A number of conclusions can be drawn immediately from the above definition.
★ The relation ''m''''ii'' = 1 means that (''r''''i'')2 = 1 for all ''i'' ; the generators are involutions.
★ If ''m''''ij'' = 2, then the generators ''r''''i'' and ''r''''j'' commute. This follows by observing that
::''xx'' = ''yy'' = 1,
: together with
:: ''xyxy'' = 1
: implies that
:: ''xy'' = ''xxyxyy'' = ''yx''.
★ In order to avoid redundancy among the relations, it is necessary to assume that ''mij''=''mji''. This follows by observing that
::''yy'' = 1,
: together with
:: (''xy'')''m'' = 1
: implies that
:: (''yx'')''m'' = (''yx'')''m''''yy'' = ''y''(''xy'')''m''''y'' = ''yy'' = 1.
The 'Coxeter matrix' is the ''n''×''n'', symmetric matrix with entries ''mij''. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal serves to define a Coxeter group.
The Coxeter matrix can be conveniently encoded by a 'Coxeter graph', as per the following rules.
★ The vertices of the graph are labelled by generator subscripts.
★ Vertices ''i'' and ''j'' are connected if and only if ''m''''ij'' ≥ 3.
★ An edge is labelled with the value of ''m''''ij'' whenever it is 4 or greater.
In particular, two generators commute if and only if they are not connected by an edge.
Furthermore, if a Coxeter graph has two or more connected components, the associated group is the direct product of the groups associated to the individual components.
An example
The graph in which vertices 1 through ''n'' are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group ''S''''n''+1; the generators correspond to the transpositions (1 2), (2 3), ... (''n'' ''n''+1). Two non-consecutive transpositions always commute, while (''k'' ''k''+1) (''k''+1 ''k''+2) gives the 3-cycle (''k'' ''k''+1 ''k''+2). Of course this only shows that ''Sn+1'' is a quotient group of the Coxeter group, but it is not too difficult to check that equality holds.
Finite Coxeter groups
Coxeter graphs of the finite Coxeter groups.
Every Weyl group can be realized as a Coxeter group. The Coxeter graph can be obtained from the Dynkin diagram by replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. The example given above corresponds to the Weyl group of the root system of type ''An''. The Weyl groups include most of the finite Coxeter groups, but there are additional examples as well. The following list gives all connected Coxeter graphs giving rise to finite groups:
Comparing this with the list of simple root systems, we see that ''Bn'' and ''Cn'' give rise to the same Coxeter group. Also, ''G''2 appears to be missing, but it is present under the name ''I''2(6). The additions to the list are ''H''3, ''H''4, and the ''I''2(''p'').
Some properties of the finite Coxeter groups are given in the following table:
| Group symbol | Alternate symbol | Rank | Order | Related polytopes | Coxeter-Dynkin diagram |
|---|---|---|---|---|---|
| ''A''''n'' | ''A''''n'' | ''n'' | (''n'' + 1)! | ''n''-simplex |  CDW_dot.png  CDW_3b.png CDW_dot.png CDW_3b.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png |
| ''B''''n'' = ''C''''n'' | ''C''''n'' | ''n'' | 2''n'' ''n''! | ''n''-hypercube / ''n''-cross-polytope | CDW_dot.png  CDW_4.png CDW_dot.png CDW_3b.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png |
| ''D''''n'' | ''B''''n'' | ''n'' | 2''n''−1 ''n''! | demihypercube |  CD_dot.png  CD_3b.png  CD_downbranch-00.png CD_3b.png CD_3b.png CD_dot.png CD_3b.png CD_dot.png |
| ''I''2(''p'') | ''D''2''p'' | 2 | 2''p'' | ''p''-gon | CDW_dot.png  CDW_p.png CDW_dot.png |
| ''H''3 | ''G''3 | 3 | 120 | icosahedron / dodecahedron | CDW_dot.png  CDW_5.png CDW_dot.png CDW_3b.png CDW_dot.png |
| ''F''4 | ''F''4 | 4 | 1152 | 24-cell | CDW_dot.png CDW_3b.png CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png |
| ''H''4 | ''G''4 | 4 | 14400 | 120-cell / 600-cell | CDW_dot.png CDW_5.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png |
| ''E''6 | ''E''6 | 6 | 51840 | ''E''6 polytope | CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png |
| ''E''7 | ''E''7 | 7 | 2903040 | ''E''7 polytope | CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png |
| ''E''8 | ''E''8 | 8 | 696729600 | ''E''8 polytope | CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png |
Symmetry groups of regular polytopes
All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series ''I''2(''p''). The symmetry group of a regular ''n''-simplex is the symmetric group ''S''''n''+1, also known as the Coxeter group of type ''An''. The symmetry group of the ''n''-cube is the same as that of the ''n''-cross-polytope, namely ''BCn''. The symmetry group of the regular dodecahedron and the regular icosahedron is ''H''3. In dimension 4, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group ''F''4, while the other two have symmetry group ''H''4.
The Coxeter groups of type ''D''''n'', ''E''6, ''E''7, and ''E''8 are the symmetry groups of certain semiregular polytopes.
Affine Weyl groups
The affine Weyl groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotient group is itself a Weyl group, and the Coxeter graph is obtained from the Coxeter graph of the Weyl group by adding an additional vertex and one or two additional edges. For example, for ''n'' ≥ 2, the graph consisting of ''n''+1 vertices in a circle is obtained from ''An'' in this way, and the corresponding Coxeter group is the affine Weyl group of ''An''. For ''n'' = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
A list of the Affine Coxeter groups follows:
| Group symbol | Alternate symbol | Related uniform tessellation(s) | Coxeter-Dynkin diagram |
|---|---|---|---|
| ''A''~''n-1'' | ''P''''n'' | Triangular tiling | CD downbranch-00.png  CD downbranch-33.png  CD downbranch-open.png CD downbranch-33.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-00.png |
| ''B''~''n-1'' | ''R''''n'' | Hypercubic honeycomb | CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_3b.png CDW_dot.png CDW_4.png CDW_dot.png |
| ''C''~''n-1'' | ''S''''n'' | Demihypercubic honeycomb | CD_dot.png  CD_4.png CD_dot.png CD_3b.png CD_dot.png CD_3b.png CD_3b.png CD_downbranch-00.png CD_3b.png CD_dot.png |
| ''D''~''n-1'' | ''Q''''n'' | Demihypercubic honeycomb | CD_dot.png CD_3b.png CD_downbranch-00.png CD_3b.png CD_dot.png CD_3b.png CD_3b.png CD_downbranch-00.png CD_3b.png CD_dot.png |
| ''I''~''1'' | ''W''''2'' | apeirogon | CDW_dot.png  CDW_infin.png CDW_dot.png |
| ''H''~''2'' | ''G''''3'' | Hexagonal tiling and Triangular tiling | CDW_dot.png  CDW_6.png CDW_dot.png  CDW_3.png CDW_dot.png |
| ''F''~''4'' | ''V''''5'' | Hexadecachoric_tetracomb and Icositetrachoronic tetracomb or F4 lattice | CDW_dot.png CDW_3b.png CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png |
| ''E''~''6'' | ''T''''7'' | E6_lattice | CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD 3b.png CD dot.png |
| ''E''~''7'' | ''T''''8'' | E7_lattice | CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png |
| ''E''~''8'' | ''T''''9'' | E8_lattice | CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png |
Note the subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.
Hyperbolic Coxeter groups
There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic geometry.
Bruhat order
A choice of reflection generators gives rise to a length function ''l'' on a Coxeter group, namely the minimum number of uses of generators required to express a group element. An expression for ''v'' using ''l(v)'' generators is a ''reduced word''. For example, the permutation (13) in ''S3'' has two reduced words, (12)(23)(12) and (23)(12)(23).
Using reduced words one may define two partial orders on the Coxeter group, the ''weak Bruhat order'' and the ''strong Bruhat order''. An element ''v'' exceeds an element ''u'' in the strong Bruhat order if some (or equivalently, any) reduced word for ''v'' contains a reduced word for ''u'' as a substring, where some letters (in any position) are dropped. Whereas ''v ≥ u'' in the weak Bruhat order if some reduced word for ''v'' contains a reduced word for ''u'' as an initial segment.
For example, the permutation (1 2 3) in ''S3'' has only one reduced word, (12)(23),
so covers (12), (23) in the strong order but only covers (12) in the weak order.
References
★ Larry C Grove and Clark T. Benson, ''Finite Reflection Groups'', Graduate texts in mathematics, vol. 99, Springer, (1985)
★ James E. Humphreys, ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990)
★ Richard Kane, ''Reflection Groups and Invariant Theory'', CMS Books in Mathematics, Springer (2001)
★ Anders Björner and Francesco Brenti, ''Combinatorics of Coxeter Groups'', Graduate texts in mathematics, vol. 231, Springer, (2005)
See also
★ Artin group
★ Weyl group
★ Triangle group
★ Coxeter number
★ Complex reflection group
★ Kazhdan-Lusztig polynomial
External links
★
★ Jenn, software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators.
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