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add implementation of MatrixGroups as fp groups

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Marek Kaluba 2022-04-02 14:21:42 +02:00
parent db5d60134c
commit cb933c3f87
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9 changed files with 314 additions and 0 deletions

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@ -10,6 +10,8 @@ import Random
import OrderedCollections: OrderedSet
export Alphabet, AutomorphismGroup, FreeGroup, FreeGroup, FPGroup, FPGroupElement, SpecialAutomorphismGroup
export MatrixGroups
export alphabet, evaluate, word, gens
include("types.jl")
@ -20,6 +22,8 @@ include("autgroups.jl")
include("aut_groups/sautFn.jl")
include("aut_groups/mcg.jl")
include("matrix_groups/MatrixGroups.jl")
using .MatrixGroups
include("wl_ball.jl")
end # of module Groups

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@ -0,0 +1,17 @@
module MatrixGroups
using GroupsCore
using Groups
using KnuthBendix
using LinearAlgebra # Identity matrix
using Random # GroupsCore rand
export SpecialLinearGroup, SymplecticGroup
include("abstract.jl")
include("SLn.jl")
include("Spn.jl")
end # module

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src/matrix_groups/SLn.jl Normal file
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@ -0,0 +1,42 @@
include("eltary_matrices.jl")
struct SpecialLinearGroup{N, T, R, A, S} <: MatrixGroup{N,T}
base_ring::R
alphabet::A
gens::S
function SpecialLinearGroup{N}(base_ring) where N
S = [ElementaryMatrix{N}(i,j, one(base_ring)) for i in 1:N for j in 1:N if i≠j]
alphabet = Alphabet(S)
return new{
N,
eltype(base_ring),
typeof(base_ring),
typeof(alphabet),
typeof(S)
}(base_ring, alphabet, S)
end
end
GroupsCore.ngens(SL::SpecialLinearGroup{N}) where N = N^2 - N
Base.show(io::IO, SL::SpecialLinearGroup{N, T}) where {N, T} =
print(io, "special linear group of $N×$N matrices over $T")
function Base.show(
io::IO,
::MIME"text/plain",
sl::Groups.AbstractFPGroupElement{<:SpecialLinearGroup{N}}
) where N
Groups.normalform!(sl)
print(io, "SL{$N,$(eltype(sl))} matrix: ")
KnuthBendix.print_repr(io, word(sl), alphabet(sl))
println(io)
Base.print_array(io, matrix_repr(sl))
end
Base.show(io::IO, sl::Groups.AbstractFPGroupElement{<:SpecialLinearGroup}) =
KnuthBendix.print_repr(io, word(sl), alphabet(sl))

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src/matrix_groups/Spn.jl Normal file
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@ -0,0 +1,68 @@
include("eltary_symplectic.jl")
struct SymplecticGroup{N, T, R, A, S} <: MatrixGroup{N,T}
base_ring::R
alphabet::A
gens::S
function SymplecticGroup{N}(base_ring) where N
S = symplectic_gens(N, eltype(base_ring))
alphabet = Alphabet(S)
return new{
N,
eltype(base_ring),
typeof(base_ring),
typeof(alphabet),
typeof(S)
}(base_ring, alphabet, S)
end
end
GroupsCore.ngens(Sp::SymplecticGroup) = length(Sp.gens)
Base.show(io::IO, ::SymplecticGroup{N}) where N = print(io, "group of $N×$N symplectic matrices")
function Base.show(
io::IO,
::MIME"text/plain",
sp::Groups.AbstractFPGroupElement{<:SymplecticGroup{N}}
) where {N}
print(io, "$N×$N Symplectic matrix: ")
KnuthBendix.print_repr(io, word(sp), alphabet(sp))
println(io)
Base.print_array(io, matrix_repr(sp))
end
function symplectic_gens(N, T=Int8)
iseven(N) || throw(ArgumentError("N needs to be even!"))
n = N÷2
a_ijs = [ElementarySymplectic{N}(:A, i,j, one(T)) for (i,j) in offdiagonal_indexing(n)]
b_is = [ElementarySymplectic{N}(:B, n+i,i, one(T)) for i in 1:n]
c_ijs = [ElementarySymplectic{N}(:B, n+i,j, one(T)) for (i,j) in offdiagonal_indexing(n)]
S = [a_ijs; b_is; c_ijs]
S = [S; transpose.(S)]
return unique(S)
end
function _std_symplectic_form(m::AbstractMatrix)
r,c = size(m)
r == c || return false
iseven(r) || return false
n = r÷2
Ω = zero(m)
for i in 1:n
Ω[2i-1:2i, 2i-1:2i] .= [0 -1; 1 0]
end
return Ω
end
function issymplectic(mat::M, Ω = _std_symplectic_form(mat)) where M <: AbstractMatrix
r, c = size(mat)
return Ω == transpose(mat) * Ω * mat
end

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@ -0,0 +1,41 @@
abstract type MatrixGroup{N, T} <: Groups.AbstractFPGroup end
const MatrixGroupElement{N, T} = Groups.AbstractFPGroupElement{<:MatrixGroup{N, T}}
Base.isone(g::MatrixGroupElement{N, T}) where {N, T} = matrix_repr(g) == I
function Base.:(==)(m1::M1, m2::M2) where {M1<:MatrixGroupElement, M2<:MatrixGroupElement}
parent(m1) === parent(m2) || return false
word(m1) == word(m2) && return true
return matrix_repr(m1) == matrix_repr(m2)
end
Base.size(m::MatrixGroupElement{N}) where N = (N, N)
Base.eltype(m::MatrixGroupElement{N, T}) where {N, T} = T
# three structural assumptions about matrix groups
Groups.word(sl::MatrixGroupElement) = sl.word
Base.parent(sl::MatrixGroupElement) = sl.parent
Groups.alphabet(M::MatrixGroup) = M.alphabet
Groups.rewriting(M::MatrixGroup) = alphabet(M)
Base.hash(sl::MatrixGroupElement, h::UInt) =
hash(matrix_repr(sl), hash(parent(sl), h))
Base.getindex(sl::MatrixGroupElement, i, j) = matrix_repr(sl)[i,j]
# Base.iterate(sl::MatrixGroupElement) = iterate(sl.elts)
# Base.iterate(sl::MatrixGroupElement, state) = iterate(sl.elts, state)
function matrix_repr(m::MatrixGroupElement{N, T}) where {N, T}
isempty(word(m)) && return StaticArrays.SMatrix{N, N, T}(I)
A = alphabet(parent(m))
return prod(matrix_repr(A[l]) for l in word(m))
end
function Base.rand(
rng::Random.AbstractRNG,
rs::Random.SamplerTrivial{<:MatrixGroup},
)
Mgroup = rs[]
S = gens(Mgroup)
return prod(g -> rand(Bool) ? g : inv(g), rand(S, rand(1:30)))
end

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@ -0,0 +1,31 @@
using Groups
using StaticArrays
struct ElementaryMatrix{N, T} <: Groups.GSymbol
i::Int
j::Int
val::T
ElementaryMatrix{N}(i, j, val=1) where N =
(@assert i≠j; new{N, typeof(val)}(i, j, val))
end
function Base.show(io::IO, e::ElementaryMatrix)
print(io, 'E', Groups.subscriptify(e.i), Groups.subscriptify(e.j))
!isone(e.val) && print(io, "^$(e.val)")
end
Base.:(==)(e::ElementaryMatrix{N}, f::ElementaryMatrix{N}) where N =
e.i == f.i && e.j == f.j && e.val == f.val
Base.hash(e::ElementaryMatrix, h::UInt) =
hash(typeof(e), hash((e.i, e.j, e.val), h))
Base.inv(e::ElementaryMatrix{N}) where N =
ElementaryMatrix{N}(e.i, e.j, -e.val)
function matrix_repr(e::ElementaryMatrix{N, T}) where {N, T}
m = StaticArrays.MMatrix{N, N, T}(I)
m[e.i, e.j] = e.val
x = StaticArrays.SMatrix{N, N}(m)
return x
end

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@ -0,0 +1,85 @@
using Groups
using StaticArrays
struct ElementarySymplectic{N, T} <: Groups.GSymbol
symbol::Symbol
i::Int
j::Int
val::T
function ElementarySymplectic{N}(s::Symbol, i::Integer, j::Integer, val=1) where N
@assert s (:A, :B)
@assert iseven(N)
n = N÷2
if s === :A
@assert 1 i n && 1 j n && i j
elseif s === :B
@assert xor(1 i n, 1 j n) && xor(n < i N, n < j N)
end
return new{N, typeof(val)}(s, i, j, val)
end
end
function Base.show(io::IO, s::ElementarySymplectic)
i, j = Groups.subscriptify(s.i), Groups.subscriptify(s.j)
print(io, s.symbol, i, j)
!isone(s.val) && print(io, "^$(s.val)")
end
function Base.show(io::IO, ::MIME"text/plain", s::ElementarySymplectic)
print(io, s)
print(io, " → corresponding root: ")
print(io, Roots.Root(s))
end
_ind(s::ElementarySymplectic{N}) where N = (s.i, s.j)
_local_ind(N_half::Integer, i::Integer) = ifelse(i<=N_half, i, i-N_half)
function _dual_ind(s::ElementarySymplectic{N}) where N
if s.symbol === :A && return _ind(s)
else#if s.symbol === :B
return _dual_ind(N÷2, s.i, s.j)
end
end
function _dual_ind(N_half, i, j)
@assert i <= N_half < j || j <= N_half < i
if i <= N_half # && j > N_half
i, j = j - N_half, i + N_half
else
i, j = j + N_half, i - N_half
end
return i, j
end
function Base.:(==)(s::ElementarySymplectic{N}, t::ElementarySymplectic{M}) where {N, M}
N == M || return false
s.symbol == t.symbol || return false
s.val == t.val || return false
return _ind(t) == _ind(s) || _ind(t) == _dual_ind(s)
end
Base.hash(s::ElementarySymplectic, h::UInt) =
hash(Set([_ind(s); _dual_ind(s)]), hash(s.symbol, hash(s.val, h)))
LinearAlgebra.transpose(s::ElementarySymplectic{N}) where N =
ElementarySymplectic{N}(s.symbol, s.j, s.i, s.val)
Base.inv(s::ElementarySymplectic{N}) where N =
ElementarySymplectic{N}(s.symbol, s.i, s.j, -s.val)
function matrix_repr(s::ElementarySymplectic{N, T}) where {N, T}
@assert iseven(N)
n = div(N, 2)
m = StaticArrays.MMatrix{N, N, T}(I)
i,j = _ind(s)
m[i,j] = s.val
if s.symbol === :A
m[n+j, n+i] = -s.val
else#if s.symbol === :B
if i > n
m[j+n, i-n] = s.val
else
m[j-n, i+n] = s.val
end
end
return StaticArrays.SMatrix{N, N}(m)
end

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using Groups.MatrixGroups
@testset "Matrix Groups" begin
@testset "SL(n, )" begin
SL3Z = SpecialLinearGroup{3}(Int8)
S = gens(SL3Z); union!(S, inv.(S))
E, sizes = Groups.wlmetric_ball(S, radius=4)
@test sizes = [13, 121, 883, 5455]
@testset "GroupsCore conformance" begin
test_Group_interface(SL3Z)
g = A(rand(1:length(alphabet(SL3Z)), 10))
h = A(rand(1:length(alphabet(SL3Z)), 10))
test_GroupElement_interface(g, h)
end
end
end

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@ -25,6 +25,8 @@ include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
include("free_groups.jl")
include("fp_groups.jl")
include("matrix_groups.jl")
include("AutFn.jl")
include("AutSigma_41.jl")
include("AutSigma3.jl")