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| In [[mathematics]], a '''complex reflection group''' is a [[Group (mathematics)|group]] acting on a finite-dimensional complex vector space, that is generated by '''complex reflections''': non-trivial elements that fix a complex [[hyperplane]] in space pointwise. (Complex reflections are sometimes called '''pseudo reflections''' or '''unitary reflections''' or sometimes just reflections.)
| | Emilia Shryock is my title but you can call me anything you like. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been performing it for quite a while. Hiring is his occupation. For a while she's been in South Dakota.<br><br>Feel free to visit my site: at home std testing; [http://www.dailyroads.com/profile_info.php?ID=2363798 please click the next web page], |
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| ==Classification==
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| Any real reflection group becomes a complex reflection group if we extend the scalars from
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| '''R''' to '''C'''. In particular all [[Coxeter group]]s or [[Weyl group]]s give examples of complex reflection groups.
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| Any finite complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces. So it is sufficient to classify the irreducible complex reflection groups.
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| The finite irreducible complex reflection groups were classified by {{Harvs |txt |first=G. C. |last=Shephard |author1-link=Geoffrey Colin Shephard|first2=J. A. |last2=Todd |author2-link=J. A. Todd |year=1954}}. They found an infinite family ''G''(''m'',''p'',''n'') depending on 3 positive integer parameters (with ''p'' dividing ''m''), and 34 exceptional cases, that they numbered from 4 to 37,
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| listed below. The group
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| ''G''(''m'',''p'',''n''), of order ''m''<sup>''n''</sup>''n''!/''p'', is the semidirect product of the abelian group | |
| of order ''m''<sup>''n''</sup>/''p'' whose elements are (θ<sup>''a''<sub>1</sub></sup>,θ<sup>''a''<sub>2</sub></sup>, ...,θ<sup>''a''<sub>''n''</sub></sup>), by the symmetric group ''S''<sub>''n''</sub> acting by permutations of the coordinates, where θ is a primitive ''m''th root of unity and Σ''a''<sub>''i''</sub>≡ 0 mod ''p''; it is an index ''p'' subgroup of the [[generalized symmetric group]] <math>S(m,n).</math>
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| Special cases of ''G''(''m'',''p'',''n''):
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| *''G''(''1'',''1'',''n'') is the Coxeter group ''A''<sub>''n''−1</sub>
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| *''G''(''2'',''1'',''n'') is the Coxeter group ''B''<sub>''n''</sub> = ''C''<sub>''n''</sub>
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| *''G''(''2'',''2'',''n'') is the Coxeter group ''D''<sub>''n''</sub>
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| *''G''(''m'',''p'',''1'') is a cyclic group of order ''m''/''p''.
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| *''G''(''m'',''m'',''2'') is the Coxeter group ''I''<sub>''2''</sub>(''m'') (and the Weyl group ''G''<sub>2</sub> when ''m'' = 6).
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| *The group ''G''(''m'',''p'',''n'') acts irreducibly on '''C'''<sup>''n''</sup> except in the cases ''m''=1, ''n''>1 (symmetric group) and ''G''(2,2,2) (Klein 4 group), when '''C'''<sup>''n''</sup> splits as a sum of irreducible representations of dimensions 1 and ''n''−1.
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| *The only cases when two groups ''G''(''m'',''p'',''n'') are isomorphic as complex reflection groups are that ''G''(''ma'',''pa'',1) is isomorphic to ''G''(''mb'',''pb'',1) for any positive integers ''a'',''b''. However there are other cases when two such groups are isomorphic as abstract groups.
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| *The complex reflection group ''G''(2,2,3) is isomorphic as a complex reflection group to ''G''(1,1,4) restricted to a 3 dimensional space.
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| *The complex reflection group ''G''(3,3,2) is isomorphic as a complex reflection group to ''G''(1,1,3) restricted to a 2 dimensional space.
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| *The complex reflection group ''G''(2''p'',''p'',1) is isomorphic as a complex reflection group to ''G''(1,1,2) restricted to a 1 dimensional space.
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| ==List of irreducible complex reflection groups==
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| There are a few duplicates in the first 3 lines of this list; see the previous section for details.
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| *'''ST''' is the Shephard–Todd number of the reflection group.
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| *'''Rank''' is the dimension of the complex vector space the group acts on.
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| *'''Structure''' describes the structure of the group. The symbol * stands for a [[central product]] of two groups. For rank 2, the quotient by the (cyclic) center is the group of rotations of a tetrahedron, octahedron, or icosahedron (''T'' = Alt(4), ''O'' = Sym(4), ''I'' = Alt(5), of orders 12, 24, 60), as stated in the table. For the notation 2<sup>1+4</sup>, see [[extra special group]].
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| *'''Order''' is the number of elements of the group.
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| *'''Reflections''' describes the number of reflections: 2<sup>6</sup>4<sup>12</sup> means that there are 6 reflections of order 2 and 12 of order 4.
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| *'''Degrees''' gives the degrees of the fundamental invariants of the ring of polynomial invariants. For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6.
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| {| class="wikitable" style="margin: 1em auto; text-align: center;"
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| |-
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| ! ST
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| ! Rank
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| ! Structure and names
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| ! Order
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| ! Reflections
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| ! Degrees
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| ! Codegrees
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| |-
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| | 1
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| | ''n''−1
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| | [[Symmetric group]] ''G''(1,1,''n'') = Sym(''n'')
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| | ''n''!
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| | 2<sup>''n''(''n'' − 1)/2</sup>
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| | 2, 3, ...,''n''
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| | 0,1,...,''n'' − 2
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| |-
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| | 2
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| | ''n''
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| | ''G''(''m'',''p'',''n'') ''m'' > 1, ''n'' > 1, ''p''<nowiki>|</nowiki>''m'' (''G''(2,2,2) is reducible)
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| | ''m''<sup>''n''</sup>''n''!/''p''
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| | 2<sup>''mn''(''n''−1)/2</sup>,''d''<sup>''n''φ(''d'')</sup> (''d''<nowiki>|</nowiki>''m''/''p'', ''d'' > 1)
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| | ''m'',2''m'',..,(''n'' − 1)''m''; ''mn''/''p''
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| | 0,''m'',..., (''n'' − 1)''m'' if ''p'' < ''m''; 0,''m'',...,(''n'' − 2)''m'', (''n'' − 1)''m'' − ''n'' if ''p'' = ''m''
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| |-
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| | 3
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| | 1
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| | Cyclic group ''G''(''m'',1,1) = '''Z'''<sub>''m''</sub>
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| | ''m''
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| | ''d''<sup>φ(''d'')</sup> (''d''<nowiki>|</nowiki>''m'', ''d'' > 1)
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| | ''m''
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| | 0
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| |-
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| | 4
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| | 2
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| | '''Z'''<sub>2</sub>.''T'' = 3[3]3
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| | 24
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| | 3<sup>8</sup>
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| | 4,6
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| | 0,2
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| |-
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| | 5
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| | 2
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| | '''Z'''<sub>6</sub>.''T'' = 3[4]3
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| | 72
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| | 3<sup>16</sup>
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| | 6,12
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| | 0,6
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| |-
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| | 6
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| | 2
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| | '''Z'''<sub>4</sub>.''T'' = 3[6]2
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| | 48
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| | 2<sup>6</sup>3<sup>8</sup>
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| | 4,12
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| | 0,8
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| |-
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| | 7
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| | 2
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| | '''Z'''<sub>12</sub>.''T'' = 〈3,3,3〉<sub>2</sub>
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| | 144
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| | 2<sup>6</sup>3<sup>16</sup>
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| | 12,12
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| | 0,12
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| |-
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| | 8
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| | 2
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| | '''Z'''<sub>4</sub>.''O'' = 4[3]4
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| | 96
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| | 2<sup>6</sup>4<sup>12</sup>
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| | 8,12
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| | 0,4
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| |-
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| | 9
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| | 2
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| | '''Z'''<sub>8</sub>.''O'' = 4[6]2
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| | 192
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| | 2<sup>18</sup>4<sup>12</sup>
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| | 8,24
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| | 0,16
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| |-
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| | 10
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| | 2
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| | '''Z'''<sub>12</sub>.''O'' = 4[4]3
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| | 288
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| | 2<sup>6</sup>3<sup>16</sup>4<sup>12</sup>
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| | 12,24
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| | 0,12
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| |-
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| | 11
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| | 2
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| | '''Z'''<sub>24</sub>.''O'' = 〈4,3,2〉<sub>12</sub>
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| | 576
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| | 2<sup>18</sup>3<sup>16</sup>4<sup>12
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| | 24,24
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| | 0,24
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| |-
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| | 12
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| | 2
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| | '''Z'''<sub>2</sub>.''O''= GL<sub>2</sub>('''F'''<sub>3</sub>)
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| | 48
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| | 2<sup>12</sup>
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| | 6,8
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| | 0,10
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| |-
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| | 13
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| | 2
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| | '''Z'''<sub>4</sub>.''O'' = 〈4,3,2〉<sub>2</sub>
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| | 96
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| | 2<sup>18</sup>
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| | 8,12
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| | 0,16
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| |-
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| | 14
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| | 2
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| | '''Z'''<sub>6</sub>.''O'' = 3[8]2
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| | 144
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| | 2<sup>12</sup>3<sup>16</sup>
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| | 6,24
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| | 0,18
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| |-
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| | 15
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| | 2
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| | '''Z'''<sub>12</sub>.''O'' = 〈4,3,2〉<sub>6</sub>
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| | 288
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| | 2<sup>18</sup>3<sup>16</sup>
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| | 12,24
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| | 0,24
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| |-
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| | 16
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| | 2
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| | '''Z'''<sub>10</sub>.''I'' = 5[3]5
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| | 600
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| | 5<sup>48</sup>
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| | 20,30
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| | 0,10
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| |-
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| | 17
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| | 2
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| | '''Z'''<sub>20</sub>.''I'' = 5[6]2
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| | 1200
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| | 2<sup>30</sup>5<sup>48</sup>
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| | 20,60
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| | 0,40
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| |-
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| | 18
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| | 2
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| | '''Z'''<sub>30</sub>.''I'' = 5[4]3
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| | 1800
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| | 3<sup>40</sup>5<sup>48</sup>
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| | 30,60
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| | 0,30
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| |-
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| | 19
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| | 2
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| | '''Z'''<sub>60</sub>.''I'' = 〈5,3,2〉<sub>30</sub>
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| | 3600
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| | 2<sup>30</sup>3<sup>40</sup>5<sup>48</sup>
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| | 60,60
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| | 0,60
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| |-
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| | 20
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| | 2
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| | '''Z'''<sub>6</sub>.''I'' = 3[5]3
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| | 360
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| | 3<sup>40</sup>
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| | 12,30
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| | 0,18
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| |-
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| | 21
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| | 2
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| | '''Z'''<sub>12</sub>.''I'' = 3[10]2
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| | 720
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| | 2<sup>30</sup>3<sup>40</sup>
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| | 12,60
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| | 0,48
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| |-
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| | 22
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| | 2
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| | '''Z'''<sub>4</sub>.''I'' = 〈5,3,2〉<sub>2</sub>
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| | 240
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| | 2<sup>30</sup>
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| | 12,20
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| | 0,28
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| |-
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| | 23
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| | 3
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| | W(H<sub>3</sub>) = '''Z'''<sub>2</sub> × PSL<sub>2</sub>(5), Coxeter
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| | 120
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| | 2<sup>15</sup>
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| | 2,6,10
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| | 0,4,8
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| |-
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| | 24
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| | 3
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| | W(J<sub>3</sub>(4)) = '''Z'''<sub>2</sub> × PSL<sub>2</sub>(7), [[Klein quartic|Klein]]
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| | 336
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| | 2<sup>21</sup>
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| | 4,6,14
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| | 0,8,10
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| |-
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| | 25
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| | 3
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| | W(L<sub>3</sub>) = W(P<sub>3</sub>) = 3<sup>1+2</sup>.SL<sub>2</sub>(3), [[Hessian group|Hessian]]
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| | 648
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| | 3<sup>24</sup>
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| | 6,9,12
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| | 0,3,6
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| |-
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| | 26
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| | 3
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| | W(M<sub>3</sub>) ='''Z'''<sub>2</sub> ×3<sup>1+2</sup>.SL<sub>2</sub>(3), [[Hessian group|Hessian]]
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| | 1296
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| | 2<sup>9</sup> 3<sup>24</sup>
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| | 6,12,18
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| | 0,6,12
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| |-
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| | 27
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| | 3
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| | W(J<sub>3</sub>(5)) = '''Z'''<sub>2</sub> ×('''Z'''<sub>3</sub>.Alt(6)), [[Valentiner group|Valentiner]]
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| | 2160
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| | 2<sup>45</sup>
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| | 6,12,30
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| | 0,18,24
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| |-
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| | 28
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| | 4
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| | W(F<sub>4</sub>) = (SL<sub>2</sub>(3)* SL<sub>2</sub>(3)).('''Z'''<sub>2</sub> × '''Z'''<sub>2</sub>) Weyl
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| | 1152
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| | 2<sup>12+12</sup>
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| | 2,6,8,12
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| | 0,4,6,10
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| |-
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| | 29
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| | 4
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| | W(N<sub>4</sub>) = ('''Z'''<sub>4</sub>*2<sup>1 + 4</sup>).Sym(5)
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| | 7680
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| | 2<sup>40</sup>
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| | 4,8,12,20
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| | 0,8,12,16
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| |-
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| | 30
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| | 4
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| | W(H<sub>4</sub>) = (SL<sub>2</sub>(5)*SL<sub>2</sub>(5)).'''Z'''<sub>2</sub> Coxeter
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| | 14400
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| | 2<sup>60</sup>
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| | 2, 12, 20,30
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| | 0,10,18,28
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| |-
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| | 31
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| | 4
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| | W(EN<sub>4</sub>) = W(O<sub>4</sub>) = ('''Z'''<sub>4</sub>*2<sup>1 + 4</sup>).Sp<sub>4</sub>(2)
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| | 46080
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| | 2<sup>60</sup>
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| | 8,12,20,24
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| | 0,12,16,28
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| |-
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| | 32
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| | 4
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| | W(L<sub>4</sub>) = '''Z'''<sub>3</sub> × Sp<sub>4</sub>(3)
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| | 155520
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| | 3<sup>80</sup>
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| | 12,18,24,30
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| | 0,6,12,18
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| |-
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| | 33
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| | 5
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| | W(K<sub>5</sub>) = '''Z'''<sub>2</sub> ×Ω<sub>5</sub>(3) = '''Z'''<sub>2</sub> × PSp<sub>4</sub>(3) = '''Z'''<sub>2</sub> × PSU<sub>4</sub>(2)
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| | 51840
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| | 2<sup>45</sup>
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| | 4,6,10,12,18
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| | 0,6,8,12,14
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| |-
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| | 34
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| | 6
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| | W(K<sub>6</sub>)= '''Z'''<sub>3</sub>.Ω{{su|p=−|b=6}}(3).'''Z'''<sub>2</sub>, [[Mitchell's group]]
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| | 39191040
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| | 2<sup>126</sup>
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| | 6,12,18,24,30,42
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| | 0,12,18,24,30,36
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| |-
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| | 35
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| | 6
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| | W(E<sub>6</sub>) = SO<sub>5</sub>(3) = O{{su|p=−|b=6}}</sub>(2) = PSp<sub>4</sub>(3).'''Z'''<sub>2</sub> = PSU<sub>4</sub>(2).'''Z'''<sub>2</sub>, Weyl
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| | 51840
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| | 2<sup>36</sup>
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| | 2,5,6,8,9,12
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| | 0,3,4,6,7,10
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| |-
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| | 36
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| | 7
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| | W(E<sub>7</sub>) = '''Z'''<sub>2</sub> ×Sp<sub>6</sub>(2), Weyl
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| | 2903040
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| | 2<sup>63</sup>
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| | 2,6,8,10,12,14,18
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| | 0,4,6,8,10,12,16
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| |-
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| | 37
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| | 8
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| | W(E<sub>8</sub>)= '''Z'''<sub>2</sub>.O{{su|p=+|b=8}}(2), Weyl
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| | 696729600
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| | 2<sup>120</sup>
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| | 2,8,12,14,18,20,24,30
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| | 0,6,10,12,16,18,22,28
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| |}
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| For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in {{harvs | last1=Broué | first1=Michel | last2=Malle | first2=Gunter | last3=Rouquier | first3=Raphaël | title=Complex reflection groups, braid groups, Hecke algebras | url=http://citeseer.ist.psu.edu/cache/papers/cs/14118/http:zSzzSzwww.math.jussieu.frzSz~rouquierzSzpreprintszSzbrmaro.pdf/complex-reflection-groups-braid.pdf | mr=1637497 | year=1998 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=500 | pages=127–190}}.
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| ==Degrees==
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| Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring ([[Chevalley–Shephard–Todd theorem]]). For <math>\ell</math> being the ''rank'' of the reflection group, the degrees <math>d_1 \leq d_2 \leq \ldots \leq d_\ell</math> of the generators of the ring of invariants are called ''degrees of W'' and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows:
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| *The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees.
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| *The order of a complex reflection group is the product of its degrees.
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| *The number of reflections is the sum of the degrees minus the rank.
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| *An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2.
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| *The degrees ''d''<sub>i</sub> satisfy the formula <math>\prod_{i=1}^\ell(q+d_i-1)= \sum_{w\in W}q^{\dim(V^w)}.</math>
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| ==Codegrees==
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| For <math>\ell</math> being the ''rank'' of the reflection group, the codegrees <math>d^*_1 \geq d^*_2 \geq \ldots \geq d^*_\ell</math> of W can be defined by
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| <math>\prod_{i=1}^\ell(q-d^*_i-1)= \sum_{w\in W}\det(w)q^{\dim(V^w)}.</math>
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| *For a real reflection group, the codegrees are the degrees minus 2.
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| *The number of reflection hyperplanes is the sum of the codegrees plus the rank.
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| ==Well-generated complex reflection groups==
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| An irreducible complex reflection group of ''rank'' <math>\ell</math> is generated by <math>\ell</math> or by <math>\ell+1</math> reflections. It is said to be ''well-generated'' if it is generated by <math>\ell</math> reflections ; it is proved that this is equivalent to the property <math>d_i + d^*_i = d_\ell</math> for all <math>1 \leq i \leq \ell</math>. For irreducible well-generated complex reflection groups, the Coxeter number <math>h</math> is defined to be the largest degree, <math>h := d_\ell</math>. A reducible complex reflection group is said to be well-generated if it is a product of irreducible well-generated complex reflection groups. Any finite real reflection group is well-generated.
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| ==References==
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| *{{Citation | last1=Broué | first1=Michel | last2=Malle | first2=Gunter | last3=Rouquier | first3=Raphaël | authorlink3=Raphaël Rouquier | title=Representations of groups (Banff, AB, 1994) | url=http://www.maths.ox.ac.uk/~rouquier/papers/banff.pdf | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=CMS Conf. Proc. | mr=1357192 | year=1995 | volume=16 | chapter=On complex reflection groups and their associated braid groups | pages=1–13}}
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| *{{Citation | last1=Broué | first1=Michel | last2=Malle | first2=Gunter | last3=Rouquier | first3=Raphaël | authorlink3=Raphaël Rouquier | title=Complex reflection groups, braid groups, Hecke algebras | id = {{citeseerx|10.1.1.128.2907}} | mr=1637497 | year=1998 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=500 | pages=127–190}}
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| *{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Les immeubles des groupes de tresses généralisés | doi=10.1007/BF01406236 | mr=0422673 | year=1972 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=17 | pages=273–302 | issue=4}}
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| *Hiller, Howard ''Geometry of Coxeter groups.'' Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN 0-273-08517-4*
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| *{{Citation | last1=Lehrer | first1=Gustav I. | last2=Taylor | first2=Donald E. | title=Unitary reflection groups | publisher=[[Cambridge University Press]] | series=Australian Mathematical Society Lecture Series | isbn=978-0-521-74989-3 | mr=2542964 | year=2009 | volume=20}}
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| *{{Citation | last1=Shephard | first1=G. C. | last2=Todd | first2=J. A. | title=Finite unitary reflection groups | url=http://books.google.com/?id=Bi7EKLHppuYC | mr=0059914 | year=1954 | journal=Canadian Journal of Mathematics. Journal Canadien de Mathématiques | issn=0008-414X | volume=6 | pages=274–304 | publisher=Canadian Mathematical Society | doi=10.4153/CJM-1954-028-3}}
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| ==External links==
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| *[http://magma.maths.usyd.edu.au/magma/htmlhelp/text1038.htm ''MAGMA Computational Algebra System'' page]
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| [[Category:Lie groups]]
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| [[Category:Geometry]]
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| [[Category:Group theory]]
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