In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.
Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra).
A (complex) reflection r (sometimes also called pseudo reflection or unitary reflection) of a finite-dimensional complex vector space V is an element
r\inGL(V)
\operatorname{Fix}(r):=\operatorname{ker}(r-\operatorname{Id}V)
A (finite) complex reflection group
W\subseteqGL(V)
GL(V)
Any real reflection group becomes a complex reflection group if we extend the scalars from R to C. In particular, all finite Coxeter groups or Weyl groups give examples of complex reflection groups.
A complex reflection group W is irreducible if the only W-invariant proper subspace of the corresponding vector space is the origin. In this case, the dimension of the vector space is called the rank of W.
The Coxeter number
h
n
h=
|l{R | |
|+|l{A}|}{n} |
l{R}
l{A}
Any complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces.[1] So it is sufficient to classify the irreducible complex reflection groups.
The irreducible complex reflection groups were classified by . They proved that every irreducible belonged to an infinite family G(m, p, n) depending on 3 positive integer parameters (with p dividing m) or was one of 34 exceptional cases, which they numbered from 4 to 37.[2] The group G(m, 1, n) is the generalized symmetric group; equivalently, it is the wreath product of the symmetric group Sym(n) by a cyclic group of order m. As a matrix group, its elements may be realized as monomial matrices whose nonzero elements are mth roots of unity.
The group G(m, p, n) is an index-p subgroup of G(m, 1, n). G(m, p, n) is of order mnn!/p. As matrices, it may be realized as the subset in which the product of the nonzero entries is an (m/p)th root of unity (rather than just an mth root). Algebraically, G(m, p, n) is a semidirect product of an abelian group of order mn/p by the symmetric group Sym(n); the elements of the abelian group are of the form (θa1, θa2, ..., θan), where θ is a primitive mth root of unity and Σai ≡ 0 mod p, and Sym(n) acts by permutations of the coordinates.[3]
The group G(m,p,n) acts irreducibly on Cn except in the cases m = 1, n > 1 (the symmetric group) and G(2, 2, 2) (the Klein four-group). In these cases, Cn splits as a sum of irreducible representations of dimensions 1 and n − 1.
When m = 2, the representation described in the previous section consists of matrices with real entries, and hence in these cases G(m,p,n) is a finite Coxeter group. In particular:[4]
In addition, when m = p and n = 2, the group G(p, p, 2) is the dihedral group of order 2p; as a Coxeter group, type I2(p) = [''p''] = (and the Weyl group G2 when p = 6).
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 (and both are isomorphic to the cyclic group of order m/p). However, there are other cases when two such groups are isomorphic as abstract groups.
The groups G(3, 3, 2) and G(1, 1, 3) are isomorphic to the symmetric group Sym(3). The groups G(2, 2, 3) and G(1, 1, 4) are isomorphic to the symmetric group Sym(4). Both G(2, 1, 2) and G(4, 4, 2) are isomorphic to the dihedral group of order 8. And the groups G(2p, p, 1) are cyclic of order 2, as is G(1, 1, 2).
There are a few duplicates in the first 3 lines of this list; see the previous section for details.
ST | Rank | Structure and names | Coxeter names | data-sort-type="number" | Order | Reflections | Degrees | Codegrees | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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1 | n-1 | Symmetric group G(1,1,n) = Sym(n) | n! | 2n(n - 1)/2 | 2, 3, ...,n | 0,1,...,n - 2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
2 | n | G(m,p,n) m > 1, n > 1, p|m (G(2,2,2) is reducible)||| mnn!/p| 2mn(n-1)/2,dnφ(d) (d|m/p, d > 1)| m,2m,..,(n - 1)m; mn/p| 0,m,..., (n - 1)m if p < m; 0,m,...,(n - 2)m, (n - 1)m - n if p = m|-| 2| 2| G(p,1,2) p > 1, || p[4]2 or | 2p2| 2p,d2φ(d) (d|p, d > 1)| p; 2p| 0,p|-| 2| 2| Dihedral group G(p,p,2) p > 2 || [''p''] or | 2p| 2p| 2,p| 0,p-2|-| 3| 1| Cyclic group G(p,1,1) = Zp|| p[] or | p| dφ(d) (d|p, d > 1)| p| 0|-| 4| 2| W(L2), Z2.T || 3[3]3 or, ⟨2,3,3⟩| 24| 38| 4,6| 0,2|-| 5| 2| Z6.T || 3[4]3 or | 72| 316| 6,12| 0,6|-| 6| 2| Z4.T || 3[6]2 or | 48| 2638| 4,12| 0,8|-| 7| 2| Z12.T ||‹3,3,3›2 or ⟨2,3,3⟩6| 144| 26316| 12,12| 0,12|-| 8| 2| Z4.O || 4[3]4 or | 96| 26412| 8,12| 0,4|-| 9| 2| Z8.O || 4[6]2 or or ⟨2,3,4⟩4| 192| 218412| 8,24| 0,16|-| 10| 2| Z12.O|| 4[4]3 or | 288| 26316412| 12,24| 0,12|-| 11| 2| Z24.O ||⟨2,3,4⟩12| 576| 218316412| 24,24| 0,24|-| 12| 2| Z2.O= GL2(F3)|| ⟨2,3,4⟩| 48| 212| 6,8| 0,10|-| 13| 2| Z4.O ||⟨2,3,4⟩2| 96| 218| 8,12| 0,16|-| 14| 2| Z6.O || 3[8]2 or | 144| 212316| 6,24| 0,18|-| 15| 2| Z12.O||⟨2,3,4⟩6| 288| 218316| 12,24| 0,24|-| 16| 2| Z10.I, ⟨2,3,5⟩×Z5 || 5[3]5 or | 600| 548| 20,30| 0,10|-| 17| 2| Z20.I || 5[6]2 or | 1200| 230548| 20,60| 0,40|-| 18| 2| Z30.I || 5[4]3 or | 1800| 340548| 30,60| 0,30|-| 19| 2| Z60.I ||⟨2,3,5⟩30| 3600| 230340548| 60,60| 0,60|-| 20| 2| Z6.I || 3[5]3 or | 360| 340| 12,30| 0,18|-| 21| 2| Z12.I || 3[10]2 or | 720| 230340| 12,60| 0,48|-| 22| 2| Z4.I ||⟨2,3,5⟩2| 240| 230| 12,20| 0,28|- BGCOLOR="#f0ffff"| 23| 3| W(H3) = Z2 × PSL2(5)|| [5,3], | 120| 215| 2,6,10| 0,4,8|-| 24| 3| W(J3(4)) = Z2 × PSL2(7), Klein||[1 1 1<sup>4</sup>]4, | 336| 221| 4,6,14| 0,8,10|-| 25| 3| W(L3) = W(P3) = 31+2.SL2(3) Hessian|| 3[3]3[3]3, | 648| 324| 6,9,12| 0,3,6|-| 26| 3| W(M3) =Z2 ×31+2.SL2(3) Hessian||2[4]3[3]3, | 1296| 29 324| 6,12,18| 0,6,12|-| 27| 3| W(J3(5)) = Z2 ×(Z3.Alt(6)), Valentiner||[1 1 1<sup>5</sup>]4, [1 1 1<sup>4</sup>]5, | 2160| 245| 6,12,30| 0,18,24|- BGCOLOR="#f0ffff"| 28| 4| W(F4) = (SL2(3)* SL2(3)).(Z2 × Z2)|| [3,4,3], | 1152| 212+12| 2,6,8,12| 0,4,6,10|-| 29| 4| W(N4) = (Z4*21 + 4).Sym(5)||[1 1 2]4, | 7680| 240| 4,8,12,20| 0,8,12,16|- BGCOLOR="#f0ffff"| 30| 4| W(H4) = (SL2(5)*SL2(5)).Z2|| [5,3,3], | 14400| 260| 2,12,20,30| 0,10,18,28|-| 31| 4| W(EN4) = W(O4) = (Z4*21 + 4).Sp4(2)||| 46080 | 260| 8,12,20,24| 0,12,16,28|-| 32| 4| W(L4) = Z3 × Sp4(3)||3[3]3[3]3[3]3, | 155520| 380| 12,18,24,30| 0,6,12,18|-| 33| 5| W(K5) = Z2 ×Ω5(3) = Z2 × PSp4(3)= Z2 × PSU4(2)||[1 2 2]3, | 51840| 245| 4,6,10,12,18| 0,6,8,12,14|-| 34| 6| W(K6)= Z3.Ω(3).Z2, Mitchell's group||[1 2 3]3, | 39191040| 2126| 6,12,18,24,30,42| 0,12,18,24,30,36|- BGCOLOR="#f0ffff"| 35| 6| W(E6) = SO5(3) = O(2) = PSp4(3).Z2 = PSU4(2).Z2||[3<sup>2,2,1</sup>], | 51840| 236| 2,5,6,8,9,12| 0,3,4,6,7,10|- BGCOLOR="#f0ffff"| 36| 7| W(E7) = Z2 ×Sp6(2)||[3<sup>3,2,1</sup>], | 2903040| 263| 2,6,8,10,12,14,18| 0,4,6,8,10,12,16|- BGCOLOR="#f0ffff"| 37| 8| W(E8)= Z2.O(2)||[3<sup>4,2,1</sup>], | 696729600| 2120| 2,8,12,14,18,20,24,30| 0,6,10,12,16,18,22,28|} For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in . DegreesShephard 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 \ell d1\leqd2\leq\ldots\leqd\ell
\sumw\in
. CodegreesFor \ell
\geq
\geq\ldots\geq
\sumw\in
.
Well-generated complex reflection groupsBy definition, every complex reflection group is generated by its reflections. The set of reflections is not a minimal generating set, however, and every irreducible complex reflection groups of rank has a minimal generating set consisting of either or reflections. In the former case, the group is said to be well-generated. The property of being well-generated is equivalent to the condition di+
=d\ell 1\leqi\leq\ell For irreducible well-generated complex reflection groups, the Coxeter number defined above equals the largest degree, h=d\ell Shephard groupsThe well-generated complex reflection groups include a subset called the Shephard groups. These groups are the symmetry groups of regular complex polytopes. In particular, they include the symmetry groups of regular real polyhedra. The Shephard groups may be characterized as the complex reflection groups that admit a "Coxeter-like" presentation with a linear diagram. That is, a Shephard group has associated positive integers and such that there is a generating set satisfying the relations
=1 sisj=sjsi |i-j|>1 sisisisi … =sisisisi … This information is sometimes collected in the Coxeter-type symbol, as seen in the table above. Among groups in the infinite family, the Shephard groups are those in which . There are also 18 exceptional Shephard groups, of which three are real.[5] [6] Cartan matricesAn extended Cartan matrix defines the unitary group. Shephard groups of rank n group have n generators.Ordinary Cartan matrices have diagonal elements 2, while unitary reflections do not have this restriction.[7] For example, the rank 1 group of order p (with symbols p[],) is defined by the matrix \left[1-e2\pi\right] Given: \zetap=e2\pi,\omega=\zeta3=e2\pi=\tfrac{1}{2}(-1+i\sqrt{3}),\zeta4=e2\pi=i,\zeta5=e2\pi=\tfrac{1}{4}(\left(\sqrt5-1\right)+i\sqrt{2(5+\sqrt5)}),\tau=\tfrac{1+\sqrt5}{2},λ=\tfrac{1+i\sqrt7}{2},\omega=\tfrac{-1+i\sqrt3}{2}
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