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add graded-by-root-system Adj

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Marek Kaluba 2022-11-07 16:13:39 +01:00
parent 4b8efd2a40
commit 147211ea7a
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3 changed files with 267 additions and 0 deletions

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@ -19,6 +19,10 @@ include("certify.jl")
include("sqadjop.jl") include("sqadjop.jl")
include("roots.jl")
import .Roots
include("gradings.jl")
include("1712.07167.jl") include("1712.07167.jl")
include("1812.03456.jl") include("1812.03456.jl")

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src/gradings.jl Normal file
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## something about roots
Roots.Root(e::MatrixGroups.ElementaryMatrix{N}) where {N} =
Roots.𝕖(N, e.i) - Roots.𝕖(N, e.j)
function Roots.Root(s::MatrixGroups.ElementarySymplectic{N}) where {N}
if s.symbol === :A
return Roots.𝕖(N ÷ 2, s.i) - Roots.𝕖(N ÷ 2, s.j)
else#if s.symbol === :B
n = N ÷ 2
i, j = ifelse(s.i <= n, s.i, s.i - n), ifelse(s.j <= n, s.j, s.j - n)
return (-1)^(s.i > s.j) * (Roots.𝕖(n, i) + Roots.𝕖(n, j))
end
end
function Roots.positive(
generating_set::AbstractVector{<:MatrixGroups.ElementarySymplectic},
)
r = Roots._positive_direction(Roots.Root(first(generating_set)))
pos_gens = [
s for s in generating_set if s.val > 0.0 && dot(Roots.Root(s), r) 0.0
]
return pos_gens
end
grading(s::MatrixGroups.ElementarySymplectic) = Roots.Root(s)
grading(e::MatrixGroups.ElementaryMatrix) = Roots.Root(e)
grading(t::Groups.Transvection) = grading(Groups._abelianize(t))
function grading(g::FPGroupElement)
if length(word(g)) == 1
A = alphabet(parent(g))
return grading(A[first(word(g))])
else
throw("Grading is implemented only for generators")
end
end
_groupby(f, iter::AbstractVector) = _groupby(f.(iter), iter)
function _groupby(keys::AbstractVector{K}, vals::AbstractVector{V}) where {K,V}
@assert length(keys) == length(vals)
d = Dict(k => V[] for k in keys)
for (k, v) in zip(keys, vals)
push!(d[k], v)
end
return d
end
function laplacians(RG::StarAlgebra, S, grading)
d = _groupby(grading, S)
Δs = Dict(α => RG(length(Sα)) - sum(RG(s) for s in Sα) for (α, Sα) in d)
return Δs
end
function Adj(rootsystem::AbstractDict, subtype::Symbol)
roots = let W = mapreduce(collect, union, keys(rootsystem))
W = union!(W, -1 .* W)
end
return reduce(
+,
(
Δα * Δβ for (α, Δα) in rootsystem for (β, Δβ) in rootsystem if
PropertyT_new.Roots.classify_sub_root_system(
roots,
first(α),
first(β),
) == subtype
),
init=zero(first(values(rootsystem))),
)
end
function level(rootsystem, level::Integer)
1 level 4 || throw("level is implemented only for i ∈{1,2,3,4}")
level == 1 && return Adj(rootsystem, :C₁) # always positive
level == 2 && return Adj(rootsystem, :A₁) + Adj(rootsystem, Symbol("C₁×C₁")) + Adj(rootsystem, :C₂) # C₂ is not positive
level == 3 && return Adj(rootsystem, :A₂) + Adj(rootsystem, Symbol("A₁×C₁"))
level == 4 && return Adj(rootsystem, Symbol("A₁×A₁")) # positive
end

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module Roots
using StaticArrays
using LinearAlgebra
export Root, isproportional, isorthogonal, ~,
abstract type AbstractRoot{N,T} end
struct Root{N,T} <: AbstractRoot{N,T}
coord::SVector{N,T}
end
Root(a) = Root(SVector(a...))
function Base.:(==)(r::Root{N}, s::Root{M}) where {M,N}
M == N || return false
r.coord == s.coord || return false
return true
end
Base.hash(r::Root, h::UInt) = hash(r.coord, hash(Root, h))
Base.:+(r::Root{N,T}, s::Root{N,T}) where {N,T} = Root{N,T}(r.coord + s.coord)
Base.:-(r::Root{N,T}, s::Root{N,T}) where {N,T} = Root{N,T}(r.coord - s.coord)
Base.:-(r::Root{N}) where {N} = Root(-r.coord)
Base.:*(a::Number, r::Root) = Root(a * r.coord)
Base.:*(r::Root, a::Number) = a * r
Base.length(r::AbstractRoot) = norm(r, 2)
LinearAlgebra.norm(r::Root, p::Real=2) = norm(r.coord, p)
LinearAlgebra.dot(r::Root, s::Root) = dot(r.coord, s.coord)
cos_angle(a, b) = dot(a, b) / (norm(a) * norm(b))
function isproportional(α::AbstractRoot{N}, β::AbstractRoot{M}) where {N,M}
N == M || return false
val = abs(cos_angle(α, β))
return isapprox(val, one(val), atol=eps(one(val)))
end
function isorthogonal(α::AbstractRoot{N}, β::AbstractRoot{M}) where {N,M}
N == M || return false
val = cos_angle(α, β)
return isapprox(val, zero(val), atol=eps(one(val)))
end
function _positive_direction(α::Root{N}) where {N}
last = -1 / 2^(N - 1)
return Root{N,Float64}(
SVector(ntuple(i -> ifelse(i == N, last, (2)^-i), N)),
)
end
function positive(roots::AbstractVector{<:Root{N}}) where {N}
# return those roots for which dot(α, Root([½, ¼, …])) > 0.0
pd = _positive_direction(first(roots))
return filter(α -> dot(α, pd) > 0.0, roots)
end
Base.:~(α::AbstractRoot, β::AbstractRoot) = isproportional(α, β)
(α::AbstractRoot, β::AbstractRoot) = isorthogonal(α, β)
function Base.show(io::IO, r::Root{N}) where {N}
print(io, "Root$(r.coord)")
end
function Base.show(io::IO, ::MIME"text/plain", r::Root{N}) where {N}
lngth² = sum(x -> x^2, r.coord)
l = isinteger(sqrt(lngth²)) ? "$(sqrt(lngth²))" : "$(lngth²)"
print(io, "Root in ^$N of length $l\n", r.coord)
end
E(N, i::Integer) = Root(ntuple(k -> k == i ? 1 : 0, N))
𝕖(N, i) = E(N, i)
𝕆(N, ::Type{T}) where {T} = Root(ntuple(_ -> zero(T), N))
"""
classify_root_system(α, β)
Return the symbol of smallest system generated by roots `α` and `β`.
The classification is based only on roots length and
proportionality/orthogonality.
"""
function classify_root_system(α::AbstractRoot, β::AbstractRoot)
lα, = length(α), length(β)
if isproportional(α, β)
if lα 2
return :A₁
elseif lα 2.0
return :C₁
else
error("Unknown root system ⟨α, β⟩:\n α = $α\n β = ")
end
elseif isorthogonal(α, β)
if lα 2
return Symbol("A₁×A₁")
elseif lα 2.0
return Symbol("C₁×C₁")
elseif (lα 2.0 && 2) || (lα 2 && 2)
return Symbol("A₁×C₁")
else
error("Unknown root system ⟨α, β⟩:\n α = $α\n β = ")
end
else # ⟨α, β⟩ is 2-dimensional, but they're not orthogonal
if lα 2
return :A₂
elseif (lα 2.0 && 2) || (lα 2 && 2)
return :C₂
else
error("Unknown root system ⟨α, β⟩:\n α = $α\n β = ")
end
end
end
function proportional_root_from_system(Ω::AbstractVector{<:Root}, α::Root)
k = findfirst(v -> isproportional(α, v), Ω)
if isnothing(k)
error("Line L_α not contained in root system Ω:\n α = $α\n Ω = ")
end
return Ω[k]
end
struct Plane{R<:Root}
v1::R
v2::R
vectors::Vector{R}
end
Plane(α::R, β::R) where {R<:Root} =
Plane(α, β, [a * α + b * β for a in -3:3 for b in -3:3])
function Base.in(r::R, plane::Plane{R}) where {R}
return any(isproportional(r, v) for v in plane.vectors)
end
function classify_sub_root_system(
Ω::AbstractVector{<:Root{N}},
α::Root{N},
β::Root{N},
) where {N}
v = proportional_root_from_system(Ω, α)
w = proportional_root_from_system(Ω, β)
subsystem = filter(ω -> ω in Plane(v, w), Ω)
@assert length(subsystem) > 0
subsystem = positive(union(subsystem, -1 .* subsystem))
l = length(subsystem)
if l == 1
x = first(subsystem)
return classify_root_system(x, x)
elseif l == 2
return classify_root_system(subsystem...)
elseif l == 3
a = classify_root_system(subsystem[1], subsystem[2])
b = classify_root_system(subsystem[2], subsystem[3])
c = classify_root_system(subsystem[1], subsystem[3])
if a == b == c # it's only A₂
return a
end
C = (:C₂, Symbol("C₁×C₁"))
if (a C && b C && c C) && (:C₂ (a, b, c))
return :C₂
end
elseif l == 4
for i = 1:l
for j = (i+1):l
T = classify_root_system(subsystem[i], subsystem[j])
T == :C₂ && return :C₂
end
end
end
@error "Unknown root subsystem generated by" α β
throw("Unknown root system: $subsystem")
end
end # of module Roots