A number of functions are provided which construct various classical groups and other groups of Lie type. The effect of these functions is to define the group in terms of a set of generating matrices.
As shown by Chevalley, for each simple Lie algebra L over the complex field and for each finite field GF(q) there is an associated matrix group L(q). In general, these groups are perfect but not simple. To obtain the simple group, it is necessary to form the quotient by the centre. Similarly, as Ree and others have shown, if the associated Coxeter graph has an automorphism, of order s say, then there will be a `twisted' version sL(q) of L(q).
Generators for the series A, C, ()^2A and ()^2B are described in
"Pairs of Generators for Matrix Groups. I" (Cayley Bulletin 3) by
Don Taylor. Generators for the series B, D and ()^2D are to appear
in the Journal of Symbolic Computation as "Matrix Generators for the
Orthogonal Groups" by Rylands and Taylor. Generators for the other
series were also provided by Rylands and Taylor.
ChevalleyGroup(s, n, K: parameters) : MonStgElt, RngIntElt, FldFin -> GrpMat
Irreducible: BoolElt Default: false
Construct a matrix group over the field K (or over GF(q)) which has the adjoint Chevalley group of Lie series s and Lie rank n as the quotient modulo scalar matrices. An exception is made for the series "2A". In this case the group 2A_n(q) is SU(n + 1, q) but in the first form of the signature K must be the field GF(q^2).The possible series are:
"A": n >= 0, the special linear group A_n(q) = SL(n + 1, q).
"B": n >= 1, the orthogonal group B_n(q) = Omega(2n + 1, q).
"C": n >= 1, the symplectic group C_n(q) = Sp(2n, q).
"D": n >= 1, the orthogonal group D_n(q) = Omega+(2n, q).
"E": n in { 6, 7, 8 }, the exceptional groups E_n(q). E_6(q) is represented as a matrix group of degree 27. It is simple unless q = 1 mod 3, in which case its centre has order 3. E_7(q) is represented as a matrix group of degree 56. It is simple unless q = 1 mod 2, in which case its centre has order 2. As yet, E_8(q) is not supported.
"F": n = 4, the exceptional group F_4(q) represented as a matrix group of degree 26. If q = 3^k then this representation is reducible. An irreducible representation is not yet available.
"G": n = 2, the exceptional group G_2(q) represented as a matrix group of degree 7. If q = 2^k then this representation is reducible. An irreducible representation of degree 6 can be obtained by setting the parameter Irreducible := true.
"2A": n >= 1, K = GF(q^2), the special unitary group 2A_n(q) = SU(n + 1, q).
"2B": n = 2, q = 2^(2k + 1), the Suzuki group 2B_2(q) = Sz(q).
"2D": n >= 1, the orthogonal group 2D_n(q) = Omega-(2n, q).
"3D": n = 4, q = p^(3k), the simple exceptional group 3D_4(q).
"2E": n = 6, q = p^(2k), the simple exceptional group 2E_6(q).
"2F": n = 4, q = 2^(2k + 1), the exceptional group 2F_4(q), simple except when q = 2 when the derived group is simple and is returned by the function TitsGroup.
"2G": n = 2, K = GF(q), q = 3^(2k + 1), the exceptional group 2G_2(q), simple except when q = 3.
Magma offers several functions to construct the classical groups. For most of these functions, it is possible to specify the particular group by giving one of the following combinations of arguments:
Construct the general linear group GL(n, K), where K = GF(q) and V is an n-dimensional vector space over K.
Construct the special linear group SL(n, K), where K = GF(q) and V is an n-dimensional vector space over K.
Construct the general unitary group GU(n, K) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power.
Construct the special unitary group SU(n, K) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power.
Construct the symplectic group Sp(n, q), where K = GF(q), V is an n-dimensional vector space over K, and n is an even integer greater than or equal to 4.
Construct the general orthogonal group GO(n, K). corresponding to the n-dimensional vector space V over the field K = GF(q), where n is odd, n >= 3 and q is a prime power.
Construct the general orthogonal group GO^ + (n, K). corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
Construct the general orthogonal group GO^ - (n, K). corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
Construct the special orthogonal group SO(n, K). corresponding to the n-dimensional vector space V over the field K = GF(q), where n is odd, n >= 3 and q is a prime power.
Construct the special orthogonal group SO^ + (n, K). corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
Construct the special orthogonal group SO^ - (n, K). corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
Construct the orthogonal group Omega(n, K) (which is the kernel of the spinor norm map on SO(n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is odd, n >= 3 and q is a prime power.
Construct the orthogonal group Omega^ + (n, K) (which is the kernel of the spinor norm map on SO^ + (n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
Construct the orthogonal group Omega^ - (n, K) (which is the kernel of the spinor norm map on SO^ - (n, K)) corresponding to the n-dimensional vector space V over the field K = GF(q), where n is even, n >= 2 and q is a prime power.
The Suzuki groups are specified slightly differently, as the degree of the group is always four. Thus for this family of groups, the possible combinations of arguments are:
Construct the Suzuki simple group Sz(q), where q is of the form 2^(2n + 1). If K is given, its cardinality is q. If V is given, it must be 4-dimensional, and over K.
> F<u> := FiniteField(8); > G := SymplecticGroup(10, F); > G; MatrixGroup(10, GF(2, 3)) Generators: [ u 0 0 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0] [ 0 0 0 1 0 0 0 0 0 0] [ 0 0 0 0 u 0 0 0 0 0] [ 0 0 0 0 0 u 0 0 0 0] [ 0 0 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 0 0 u^6]
[0 0 0 1 1 1 0 0 0 0] [1 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0] [0 0 0 0 1 0 1 0 0 0] [0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0]
> F<w> := FiniteField(128); > V := VectorSpace(F, 4); > S := SuzukiGroup(V); > S; MatrixGroup(4, GF(2, 7)) Generators: [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 0 0]
[ w^8 0 0 0] [ 0 w^120 0 0] [ 0 0 w^7 0] [ 0 0 0 w^119]
[ 1 0 0 0] [ w^8 1 0 0] [ 0 w 1 0] [w^17 w^9 w^8 1] > Order(S); 34093383680 > FactoredOrder(S); [ <2, 14>, <5, 1>, <29, 1>, <113, 1>, <127, 1> ]
Given two matrix groups G and H of degrees m and n respectively, construct the direct product of G and H as a matrix group of degree m + n.
Given a sequence Q of n matrix groups, construct the direct product Q[1] x Q[2] x ... x Q[n] as a matrix group of degree equal to the sum of the degrees of the groups Q[i], (i = 1, ..., n).
Given a matrix group G over the finite field K and a subfield S of K, construct the semilinear extension of G over the subfield S.
Given a matrix group G and a permutation group H, construct action of the wreath product on the tensor power of G by H, which is the (image of) the wreath product in its action on the tensor power (of the space that G acts on). The degree of the new group is d^k where d is the degree of G and k is the degree of H.
Given a matrix group G and a permutation group H, construct the wreath product G wreath H of G and H.
> K<w> := FiniteField(4); > G := SpecialUnitaryGroup(3, K); > D := DirectProduct(G, G); > D; MatrixGroup(6, GF(2, 2)) Generators: [ 1 w w 0 0 0] [ 0 1 w^2 0 0 0] [ 0 0 1 0 0 0] [ 0 0 0 1 0 0] [ 0 0 0 0 1 0] [ 0 0 0 0 0 1]
[w 1 1 0 0 0] [1 1 0 0 0 0] [1 0 0 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1]
[ 1 0 0 0 0 0] [ 0 1 0 0 0 0] [ 0 0 1 0 0 0] [ 0 0 0 1 w w] [ 0 0 0 0 1 w^2] [ 0 0 0 0 0 1]
[1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 w 1 1] [0 0 0 1 1 0] [0 0 0 1 0 0] > Order(D); 46656 > H := SymmetricGroup(3); > E := WreathProduct(G, H); > Degree(E); 9 > Order(E); 60466176 > F := TensorWreathProduct(G, H); > Degree(F); 27 > Order(F); 6718464