62 lines
1.6 KiB
Julia
62 lines
1.6 KiB
Julia
include("eltary_symplectic.jl")
|
||
|
||
struct SymplecticGroup{N,T,R,A,S} <: AbstractMatrixGroup{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,T}) where {N,T} = print(io, "Sp{$N,$T}")
|
||
|
||
function Base.show(io::IO, ::MIME"text/plain", ::SymplecticGroup{N}) where {N}
|
||
return print(io, "group of $N×$N symplectic matrices")
|
||
end
|
||
_offdiag_idcs(n) = ((i, j) for i in 1:n for j in 1:n if i ≠ j)
|
||
|
||
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 _offdiag_idcs(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 _offdiag_idcs(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
|
||
𝕆 = zeros(eltype(m), n, n)
|
||
𝕀 = one(eltype(m)) * LinearAlgebra.I
|
||
Ω = [𝕆 -𝕀
|
||
𝕀 𝕆]
|
||
return Ω
|
||
end
|
||
|
||
function issymplectic(mat::M, Ω=_std_symplectic_form(mat)) where {M<:AbstractMatrix}
|
||
return Ω == transpose(mat) * Ω * mat
|
||
end
|