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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,217 @@ | ||
| """ | ||
| Character <: AbstractClassFunction | ||
| Struct representing (possibly virtual) character of a group. | ||
| Characters are backed by `table(χ)::CharacterTable` which actually stores the | ||
| character values. The multiplicities (decomposition into the irreducible | ||
| summands) of a given character can be obtained by calling `multiplicities(χ)` | ||
| which returns a vector of coefficients of `χ` in the basis of | ||
| `irreducible_characters(table(χ))`. | ||
| It is assumed that equal class functions on the same group will have | ||
| **identical** (ie. `===`) character tables. | ||
| """ | ||
| struct Character{T,S,ChT<:CharacterTable} <: AbstractClassFunction{T} | ||
| table::ChT | ||
| multips::Vector{S} | ||
| end | ||
|
|
||
| function Character{R}( | ||
| chtbl::CharacterTable{Gr,T}, | ||
| multips::AbstractVector{S}, | ||
| ) where {R,Gr,T,S} | ||
| return Character{R,S,typeof(chtbl)}(chtbl, multips) | ||
| end | ||
|
|
||
| function Character(chtbl::CharacterTable, multips::AbstractVector) | ||
| R = Base._return_type(*, Tuple{eltype(chtbl),eltype(multips)}) | ||
| @assert R ≠ Any | ||
| return Character{R}(chtbl, multips) | ||
| end | ||
|
|
||
| function Character(chtbl::CharacterTable, i::Integer) | ||
| return Character{eltype(chtbl)}(chtbl, i) | ||
| end | ||
|
|
||
| function Character{T}(chtbl::CharacterTable, i::Integer) where {T} | ||
| v = zeros(Int, nconjugacy_classes(chtbl)) | ||
| v[i] = 1 | ||
| return Character{T,Int,typeof(chtbl)}(chtbl, v) | ||
| end | ||
|
|
||
| function Character{T}(χ::Character) where {T} | ||
| S = eltype(multiplicities(χ)) | ||
| ChT = typeof(table(χ)) | ||
| return Character{T,S,ChT}(table(χ), multiplicities(χ)) | ||
| end | ||
|
|
||
| ## Accessors | ||
| table(χ::Character) = χ.table | ||
| multiplicities(χ::Character) = χ.multips | ||
|
|
||
| ## AbstractClassFunction api | ||
| Base.parent(χ::Character) = parent(table(χ)) | ||
| conjugacy_classes(χ::Character) = conjugacy_classes(table(χ)) | ||
| function Base.values(χ::Character{T}) where {T} | ||
| return T[χ[i] for i in 1:nconjugacy_classes(table(χ))] | ||
| end | ||
|
|
||
| Base.@propagate_inbounds function Base.getindex( | ||
| χ::Character{T}, | ||
| i::Integer, | ||
| ) where {T} | ||
| i = i < 0 ? table(χ).inv_of[abs(i)] : i | ||
| @boundscheck 1 ≤ i ≤ nconjugacy_classes(table(χ)) | ||
|
|
||
| return convert( | ||
| T, | ||
| sum( | ||
| c * table(χ)[idx, i] for | ||
| (idx, c) in enumerate(multiplicities(χ)) if !iszero(c); | ||
| init = zero(T), | ||
| ), | ||
| ) | ||
| end | ||
|
|
||
| ## Basic functionality | ||
|
|
||
| function Base.:(==)(χ::Character, ψ::Character) | ||
| return table(χ) === table(ψ) && multiplicities(χ) == multiplicities(ψ) | ||
| end | ||
| Base.hash(χ::Character, h::UInt) = hash(table(χ), hash(multiplicities(χ), h)) | ||
|
|
||
| function Base.deepcopy_internal(χ::Character{T}, d::IdDict) where {T} | ||
| haskey(d, χ) && return d[χ] | ||
| return Character{T}(table(χ), copy(multiplicities(χ))) | ||
| end | ||
|
|
||
| ## Character arithmetic | ||
|
|
||
| for f in (:+, :-) | ||
| @eval begin | ||
| function Base.$f(χ::Character, ψ::Character) | ||
| @assert table(χ) === table(ψ) | ||
| return Character(table(χ), $f(multiplicities(χ), multiplicities(ψ))) | ||
| end | ||
| end | ||
| end | ||
|
|
||
| Base.:*(χ::Character, c::Number) = Character(table(χ), c .* multiplicities(χ)) | ||
| Base.:*(c::Number, χ::Character) = χ * c | ||
| Base.:/(χ::Character, c::Number) = Character(table(χ), multiplicities(χ) ./ c) | ||
|
|
||
| Base.zero(χ::Character) = 0 * χ | ||
|
|
||
| function __decompose(T::Type, values::AbstractVector, tbl::CharacterTable) | ||
| ψ = ClassFunction(values, conjugacy_classes(tbl), tbl.inv_of) | ||
| return decompose(T, ψ, tbl) | ||
| end | ||
|
|
||
| function decompose(cfun::AbstractClassFunction, tbl::CharacterTable) | ||
| return decompose(eltype(cfun), cfun, tbl) | ||
| end | ||
|
|
||
| function decompose(T::Type, cfun::AbstractClassFunction, tbl::CharacterTable) | ||
| vals = Vector{T}(undef, length(conjugacy_classes(tbl))) | ||
| return decompose!(vals, cfun, tbl) | ||
| end | ||
|
|
||
| function decompose!( | ||
| vals::AbstractVector, | ||
| cfun::AbstractClassFunction, | ||
| tbl::CharacterTable, | ||
| ) | ||
| @assert length(vals) == length(conjugacy_classes(tbl)) | ||
| @assert conjugacy_classes(tbl) === conjugacy_classes(cfun) | ||
|
|
||
| for (i, idx) in enumerate(eachindex(vals)) | ||
| χ = Character(tbl, i) | ||
| vals[idx] = dot(χ, cfun) | ||
| end | ||
| return vals | ||
| end | ||
|
|
||
| function Base.:*(χ::Character, ψ::Character) | ||
| @assert table(χ) == table(ψ) | ||
| values = Characters.values(χ) .* Characters.values(ψ) | ||
| return Character(table(χ), __decompose(Int, values, table(χ))) | ||
| end | ||
|
|
||
| Base.:^(χ::Character, n::Integer) = Base.power_by_squaring(χ, n) | ||
|
|
||
| ## Group-theoretic functions: | ||
|
|
||
| PermutationGroups.degree(χ::Character) = Int(χ(one(parent(χ)))) | ||
| function PermutationGroups.degree( | ||
| χ::Character{T,CCl}, | ||
| ) where {T,CCl<:AbstractOrbit{<:AbstractMatrix}} | ||
| return Int(χ[1]) | ||
| end | ||
|
|
||
| function Base.conj(χ::Character{T,S}) where {T,S} | ||
| vals = collect(values(χ)) | ||
| all(isreal, vals) && return Character{T}(χ) | ||
| tbl = table(χ) | ||
| ψ = ClassFunction(vals[tbl.inv_of], conjugacy_classes(tbl), tbl.inv_of) | ||
| multips = S[dot(ψ, χ) for χ in irreducible_characters(tbl)] | ||
| return Character{T,eltype(multips),typeof(tbl)}(tbl, multips) | ||
| end | ||
|
|
||
| function isvirtual(χ::Character) | ||
| return any(<(0), multiplicities(χ)) || any(!isinteger, multiplicities(χ)) | ||
| end | ||
|
|
||
| function isirreducible(χ::Character) | ||
| C = multiplicities(χ) | ||
| k = findfirst(!iszero, C) | ||
| k !== nothing || return false # χ is zero | ||
| isone(C[k]) || return false # muliplicity is ≠ 1 | ||
| kn = findnext(!iszero, C, k + 1) | ||
| kn === nothing && return true # there is only one ≠ 0 entry | ||
| return false | ||
| end | ||
|
|
||
| """ | ||
| affordable_real!(χ::Character) | ||
| Return either `χ` or `2re(χ)` depending whether `χ` is afforded by a real | ||
| representation, modifying `χ` in place. | ||
| """ | ||
| function affordable_real!(χ::Character) | ||
| ι = frobenius_schur(χ) | ||
| if ι <= 0 # i.e. χ is complex or quaternionic | ||
| χ.multips .+= multiplicities(conj(χ)) | ||
| end | ||
| return χ | ||
| end | ||
|
|
||
| """ | ||
| frobenius_schur(χ::AbstractClassFunction[, pmap::PowerMap]) | ||
| Return Frobenius-Schur indicator of `χ`, i.e. `Σχ(g²)` where sum is taken over | ||
| the whole group. | ||
| If χ is an irreducible `Character`, Frobenius-Schur indicator takes values in | ||
| `{1, 0, -1}` which correspond to the following situations: | ||
| 1. `χ` is real-valued and is afforded by an irreducible real representation, | ||
| 2. `χ` is a complex character which is not afforded by a real representation, and | ||
| 3. `χ` is quaternionic character, i.e. it is real valued, but is not afforded by a | ||
| real representation. | ||
| In cases 2. and 3. `2re(χ) = χ + conj(χ)` corresponds to an irreducible character | ||
| afforded by a real representation. | ||
| """ | ||
| function frobenius_schur(χ::Character) | ||
| @assert isirreducible(χ) | ||
|
|
||
| pmap = powermap(table(χ)) | ||
| ι = sum( | ||
| length(c) * χ[pmap[i, 2]] for (i, c) in enumerate(conjugacy_classes(χ)) | ||
| ) | ||
|
|
||
| ι_int = Int(ι) | ||
| ordG = sum(length, conjugacy_classes(χ)) | ||
| d, r = divrem(ι_int, ordG) | ||
| @assert r == 0 "Non integral Frobenius Schur Indicator: $(ι_int) = $d * $ordG + $r" | ||
| return d | ||
| end | ||
|
|
||
| Base.isreal(χ::Character) = frobenius_schur(χ) > 0 |
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