# Matrix groups

matrix_groupMethod
matrix_group(R::Ring, m::Int, V::T...) where T<:Union{MatElem,MatrixGroupElem}
matrix_group(R::Ring, m::Int, V::AbstractVector{T}) where T<:Union{MatElem,MatrixGroupElem}
matrix_group(V::T...) where T<:Union{MatElem,MatrixGroupElem}
matrix_group(V::AbstractVector{T}) where T<:Union{MatElem,MatrixGroupElem}

Return the matrix group generated by matrices in V. If the degree m and coefficient ring R are not given, then V must be non-empty

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MatrixGroupType
MatrixGroup{RE<:RingElem, T<:MatElem{RE}} <: GAPGroup

Type of groups G of n x n matrices over the ring R, where n = degree(G) and R = base_ring(G).

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MatrixGroupElemType
MatrixGroupElem{RE<:RingElem, T<:MatElem{RE}} <: AbstractMatrixGroupElem

Elements of a group of type MatrixGroup{RE<:RingElem, T<:MatElem{RE}}

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base_ringMethod
base_ring(G::MatrixGroup)

Return the base ring of the matrix group G.

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degreeMethod
degree(G::MatrixGroup)

Return the degree of the matrix group G, i.e. the number of rows of its matrices.

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centralizerMethod
centralizer(G::MatrixGroup{T}, x::MatrixGroupElem{T})

Return (C,f), where C is the centralizer of x in C and f is the embedding of C into G. If G = GL(n,F) or SL(n,F), then f = nothing. In this case, to get the embedding homomorphism of C into G, use

is_subgroup(C, G)[2]

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map_entriesMethod
map_entries(f, G::MatrixGroup)

Return the matrix group obtained by applying f element-wise to each generator of G.

f can be a ring or a field, a suitable map, or a Julia function.

Examples

julia> mat = matrix(ZZ, 2, 2, [1, 1, 0, 1]);

julia> G = matrix_group(mat);

julia> G2 = map_entries(x -> -x, G)
Matrix group of degree 2
over integer ring

julia> is_finite(G2)
false

julia> order(map_entries(GF(3), G))
3
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## Elements of matrix groups

matrixMethod
matrix(x::MatrixGroupElem)

Return the underlying matrix of x.

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base_ringMethod
base_ring(x::MatrixGroupElem)

Return the base ring of the underlying matrix of x.

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nrowsMethod
number_of_rows(x::MatrixGroupElem)

Return the number of rows of the underlying matrix of x.

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detMethod
det(x::MatrixGroupElem)

Return the determinant of the underlying matrix of x.

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trMethod
tr(x::MatrixGroupElem)

Return the trace of the underlying matrix of x.

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multiplicative_jordan_decompositionMethod
multiplicative_jordan_decomposition(M::MatrixGroupElem)

Return S and U in the group G = parent(M) such that S is semisimple, U is unipotent and M = SU = US.

WARNING:

this is NOT, in general, the same output returned when M has type MatElem.

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is_semisimpleMethod
is_semisimple(x::MatrixGroupElem{T}) where T <: FinFieldElem

Return whether x is semisimple, i.e. has order coprime with the characteristic of its base ring.

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is_unipotentMethod
is_unipotent(x::MatrixGroupElem{T}) where T <: FinFieldElem

Return whether x is unipotent, i.e. its order is a power of the characteristic of its base ring.

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## Sesquilinear forms

SesquilinearFormType
SesquilinearForm{T<:RingElem}

Type of groups G of n x n matrices over the ring R, where n = degree(G) and R = base_ring(G). At the moment, only rings of type fqPolyRepField are supported.

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is_alternatingMethod
is_alternating(f::SesquilinearForm)

Return whether the form f is an alternating form.

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is_hermitianMethod
is_hermitian(f::SesquilinearForm)

Return whether the form f is a hermitian form.

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is_quadraticMethod
is_quadratic(f::SesquilinearForm)

Return whether the form f is a quadratic form.

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is_symmetricMethod
is_symmetric(f::SesquilinearForm)

Return whether the form f is a symmetric form.

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quadratic_formMethod
quadratic_form(f::MPolyRingElem{T}; check=true)

Return the quadratic form described by the polynomial f. Here, f must be a homogeneous polynomial of degree 2. If check is set as false, it does not check whether the polynomial is homogeneous of degree 2. To define quadratic forms of dimension 1, f can also have type PolyRingElem{T}.

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corresponding_bilinear_formMethod
corresponding_bilinear_form(Q::SesquilinearForm)

Given a quadratic form Q, return the bilinear form B defined by B(u,v) = Q(u+v)-Q(u)-Q(v).

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corresponding_quadratic_formMethod
corresponding_quadratic_form(Q::SesquilinearForm)

Given a symmetric form f, returns the quadratic form Q defined by Q(v) = f(v,v)/2. It is defined only in odd characteristic.

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gram_matrixMethod
gram_matrix(B::SesquilinearForm)

Return the Gram matrix of a sesquilinear or quadratic form B.

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radicalMethod
radical(f::SesquilinearForm{T})

Return the radical of the sesquilinear form f, i.e. the subspace of all v such that f(u,v)=0 for all u. The radical of a quadratic form Q is the set of vectors v such that Q(v)=0 and v lies in the radical of the corresponding bilinear form.

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witt_indexMethod
witt_index(f::SesquilinearForm{T})

Return the Witt index of the form induced by f on V/Rad(f). The Witt Index is the dimension of a maximal totally isotropic (singular for quadratic forms) subspace.

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is_degenerateMethod
is_degenerate(f::SesquilinearForm{T})

Return whether f is degenerate, i.e. f has nonzero radical. A quadratic form is degenerate if the corresponding bilinear form is.

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is_singularMethod
is_singular(Q::SesquilinearForm{T})

For a quadratic form Q, return whether Q is singular, i.e. Q has nonzero radical.

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is_congruentMethod
is_congruent(f::SesquilinearForm{T}, g::SesquilinearForm{T}) where T <: RingElem

If f and g are sesquilinear forms, return (true, C) if there exists a matrix C such that f^C = g, or equivalently, CBC* = A, where A and B are the Gram matrices of f and g respectively, and C* is the transpose-conjugate matrix of C. If such C does not exist, then return (false, nothing).

If f and g are quadratic forms, return (true, C) if there exists a matrix C such that f^A = ag for some scalar a. If such C does not exist, then return (false, nothing).

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## Invariant forms

invariant_bilinear_formsMethod
invariant_bilinear_forms(G::MatrixGroup)

Return a generating set for the vector spaces of bilinear forms preserved by the group G.

Note:

At the moment, elements of the generating set are returned of type mat_elem_type(G).

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invariant_sesquilinear_formsMethod
invariant_sesquilinear_forms(G::MatrixGroup)

Return a generating set for the vector spaces of sesquilinear non-bilinear forms preserved by the group G. An exception is thrown if base_ring(G) is not a finite field with even degree over its prime subfield.

Note:

At the moment, elements of the generating set are returned of type mat_elem_type(G).

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invariant_quadratic_formsMethod
invariant_quadratic_forms(G::MatrixGroup)

Return a generating set for the vector spaces of quadratic forms preserved by the group G.

Note:

At the moment, elements of the generating set are returned of type mat_elem_type(G).

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invariant_symmetric_formsMethod
invariant_symmetric_forms(G::MatrixGroup)

Return a generating set for the vector spaces of symmetric forms preserved by the group G.

Note:

At the moment, elements of the generating set are returned of type mat_elem_type(G).

Note:

Work properly only in odd characteristic. In even characteristic, only alternating forms are found.

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invariant_alternating_formsMethod
invariant_alternating_forms(G::MatrixGroup)

Return a generating set for the vector spaces of alternating forms preserved by the group G.

Note:

At the moment, elements of the generating set are returned of type mat_elem_type(G).

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invariant_hermitian_formsMethod
invariant_hermitian_forms(G::MatrixGroup)

Return a generating set for the vector spaces of hermitian forms preserved by the group G. An exception is thrown if base_ring(G) is not a finite field with even degree over its prime subfield.

Note:

At the moment, elements of the generating set are returned of type mat_elem_type(G).

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invariant_bilinear_formMethod
invariant_bilinear_form(G::MatrixGroup)

Return an invariant bilinear form for the group G. An exception is thrown if the module induced by the action of G is not absolutely irreducible.

Note:

At the moment, the output is returned of type mat_elem_type(G).

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invariant_sesquilinear_formMethod
invariant_sesquilinear_form(G::MatrixGroup)

Return an invariant sesquilinear (non bilinear) form for the group G. An exception is thrown if the module induced by the action of G is not absolutely irreducible or if the group is defined over a finite field of odd degree over the prime field.

Note:

At the moment, the output is returned of type mat_elem_type(G).

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invariant_quadratic_formMethod
invariant_quadratic_form(G::MatrixGroup)

Return an invariant quadratic form for the group G. An exception is thrown if the module induced by the action of G is not absolutely irreducible.

Note:

At the moment, the output is returned of type mat_elem_type(G).

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preserved_quadratic_formsMethod
preserved_quadratic_forms(G::MatrixGroup)

Uses random methods to find all of the quadratic forms preserved by G up to a scalar (i.e. such that G is a group of similarities for the forms). Since the procedure relies on a pseudo-random generator, the user may need to execute the operation more than once to find all invariant quadratic forms.

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preserved_sesquilinear_formsMethod
preserved_sesquilinear_forms(G::MatrixGroup)

Uses random methods to find all of the sesquilinear forms preserved by G up to a scalar (i.e. such that G is a group of similarities for the forms). Since the procedure relies on a pseudo-random generator, the user may need to execute the operation more than once to find all invariant sesquilinear forms.

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isometry_groupMethod
isometry_group(f::SesquilinearForm{T})

Return the group of isometries for the sesquilinear form f.

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orthogonal_signMethod
orthogonal_sign(G::MatrixGroup)

For absolutely irreducible G of degree n and such that base_ring(G) is a finite field, return

• nothing if G does not preserve a nonzero quadratic form,
• 0 if n is odd and G preserves a nonzero quadratic form,
• 1 if n is even and G preserves a nonzero quadratic form of + type,
• -1 if n is even and G preserves a nonzero quadratic form of - type.
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## Utilities for matrices

pol_elementary_divisorsMethod
pol_elementary_divisors(x::MatElem)
pol_elementary_divisors(x::MatrixGroupElem)

Return a list of pairs (f_i,m_i), for irreducible polynomials f_i and positive integers m_i, where the f_i^m_i are the elementary divisors of x.

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generalized_jordan_blockMethod
generalized_jordan_block(f::T, n::Int) where T<:PolyRingElem

Return the Jordan block of dimension n corresponding to the polynomial f.

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generalized_jordan_formMethod
generalized_jordan_form(A::MatElem{T}; with_pol::Bool=false) where T

Return (J,Z), where Z^-1*J*Z = A and J is a diagonal join of Jordan blocks (corresponding to irreducible polynomials).

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matrixMethod
matrix(A::Vector{AbstractAlgebra.Generic.FreeModuleElem})

Return the matrix whose rows are the vectors in A. All vectors in A must have the same length and the same base ring.

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upper_triangular_matrixMethod
upper_triangular_matrix(L)

Return the upper triangular matrix whose entries on and above the diagonal are the elements of L.

An exception is thrown whenever the length of L is not equal to $n(n+1)/2$, for some integer $n$.

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lower_triangular_matrixMethod
lower_triangular_matrix(L)

Return the upper triangular matrix whose entries on and below the diagonal are the elements of L.

An exception is thrown whenever the length of L is not equal to $n(n+1)/2$, for some integer $n$.

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conjugate_transposeMethod
conjugate_transpose(x::MatElem{T}) where T <: FinFieldElem

If the base ring of x is GF(q^2), return the matrix transpose( map ( y -> y^q, x) ). An exception is thrown if the base ring does not have even degree.

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complementMethod
complement(V::AbstractAlgebra.Generic.FreeModule{T}, W::AbstractAlgebra.Generic.Submodule{T}) where T <: FieldElem

Return a complement for W in V, i.e. a subspace U of V such that V is direct sum of U and W.

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permutation_matrixMethod
permutation_matrix(F::Ring, Q::AbstractVector{T}) where T <: Int
permutation_matrix(F::Ring, p::PermGroupElem)

Return the permutation matrix over the ring R corresponding to the sequence Q or to the permutation p. If Q is a sequence, then Q must contain exactly once every integer from 1 to some n.

Examples

julia> s = perm([3,1,2])
(1,3,2)

julia> permutation_matrix(QQ,s)
[0   0   1]
[1   0   0]
[0   1   0]

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is_alternatingMethod
is_alternating(B::MatElem)

Return whether the form corresponding to the matrix B is alternating, i.e. B = -transpose(B) and B has zeros on the diagonal. Return false if B is not a square matrix.

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is_hermitianMethod
is_hermitian(B::MatElem{T}) where T <: FinFieldElem

Return whether the matrix B is hermitian, i.e. B = conjugate_transpose(B). Return false if B is not a square matrix, or the field has not even degree.

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## Classical groups

general_linear_groupMethod
general_linear_group(n::Int, q::Int)
general_linear_group(n::Int, R::Ring)
GL = general_linear_group

Return the general linear group of dimension n over the ring R respectively the field GF(q).

Currently, this function only supports rings of type FqField.

Examples

julia> F = GF(7,1)
Prime field of characteristic 7

julia> H = general_linear_group(2,F)
GL(2,7)

julia> gens(H)
2-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[3 0; 0 1]
[6 1; 6 0]

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special_linear_groupMethod
special_linear_group(n::Int, q::Int)
special_linear_group(n::Int, R::Ring)
SL = special_linear_group

Return the special linear group of dimension n over the ring R respectively the field GF(q).

Currently, this function only supports rings of type FqField.

Examples

julia> F = GF(7,1)
Prime field of characteristic 7

julia> H = special_linear_group(2,F)
SL(2,7)

julia> gens(H)
2-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[3 0; 0 5]
[6 1; 6 0]

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symplectic_groupMethod
symplectic_group(n::Int, q::Int)
symplectic_group(n::Int, R::Ring)
Sp = symplectic_group

Return the symplectic group of dimension n over the ring R respectively the field GF(q). The dimension n must be even.

Currently, this function only supports rings of type FqField.

Examples

julia> F = GF(7,1)
Prime field of characteristic 7

julia> H = symplectic_group(2,F)
Sp(2,7)

julia> gens(H)
2-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[3 0; 0 5]
[6 1; 6 0]

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orthogonal_groupMethod
orthogonal_group(e::Int, n::Int, R::Ring)
orthogonal_group(e::Int, n::Int, q::Int)
GO = orthogonal_group

Return the orthogonal group of dimension n over the ring R respectively the field GF(q), and of type e, where e in {+1,-1} for n even and e=0 for n odd. If n is odd, e can be omitted.

Currently, this function only supports rings of type FqField.

Examples

julia> F = GF(7,1)
Prime field of characteristic 7

julia> H = symplectic_group(2,F)
Sp(2,7)

julia> gens(H)
2-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[3 0; 0 5]
[6 1; 6 0]

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special_orthogonal_groupMethod
special_orthogonal_group(e::Int, n::Int, R::Ring)
special_orthogonal_group(e::Int, n::Int, q::Int)
SO = special_orthogonal_group

Return the special orthogonal group of dimension n over the ring R respectively the field GF(q), and of type e, where e in {+1,-1} for n even and e=0 for n odd. If n is odd, e can be omitted.

Currently, this function only supports rings of type FqField.

Examples

julia> F = GF(7,1)
Prime field of characteristic 7

julia> H = special_orthogonal_group(1,2,F)
SO+(2,7)

julia> gens(H)
3-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[3 0; 0 5]
[5 0; 0 3]
[1 0; 0 1]

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omega_groupMethod
omega_group(e::Int, n::Int, R::Ring)
omega_group(e::Int, n::Int, q::Int)

Return the Omega group of dimension n over the field GF(q) of type e, where e in {+1,-1} for n even and e=0 for n odd. If n is odd, e can be omitted.

Currently, this function only supports rings of type FqField.

Examples

julia> F = GF(7,1)
Prime field of characteristic 7

julia> H = omega_group(1,2,F)
Omega+(2,7)

julia> gens(H)
1-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[2 0; 0 4]

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unitary_groupMethod
unitary_group(n::Int, q::Int)
GU = unitary_group

Return the unitary group of dimension n over the field GF(q^2).

Examples

julia> H = unitary_group(2,3)
GU(2,3)

julia> gens(H)
2-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[o 0; 0 2*o]
[2 2*o+2; 2*o+2 0]
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special_unitary_groupMethod
special_unitary_group(n::Int, q::Int)
SU = special_unitary_group

Return the special unitary group of dimension n over the field with q^2 elements.

Examples

julia> H = special_unitary_group(2,3)
SU(2,3)

julia> gens(H)
2-element Vector{MatrixGroupElem{FqFieldElem, FqMatrix}}:
[1 2*o+2; 0 1]
[0 2*o+2; 2*o+2 0]
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