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Add letters function for PcGroupElem #4202

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Add letters function for PcGroupElem #4202

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e9a4ac4
Add syllable functions to pcgroup
jamesnohilly Oct 14, 2024
b910923
Add letters function for PcGroupElem
jamesnohilly Oct 17, 2024
d362f67
Add testset for PcGroup letters function
jamesnohilly Oct 17, 2024
27c24e7
Add support for SubPcGroupElem in letters
jamesnohilly Oct 17, 2024
89b8fe6
Cleanup code
jamesnohilly Oct 17, 2024
94cd03e
Rename pcgroup exponent_vector function
jamesnohilly Oct 18, 2024
416fd8b
Fix test cases for finite pcgroup
jamesnohilly Oct 30, 2024
c350414
Add check for syllables in canonical form
jamesnohilly Oct 31, 2024
b552be6
Add testset for canonical form check
jamesnohilly Oct 31, 2024
11bdb25
Refactor based on PR comments
jamesnohilly Nov 1, 2024
1b80584
Document pcgroup syllable function
jamesnohilly Nov 1, 2024
2d0bfb8
Improve letters and syllables documentation
jamesnohilly Nov 15, 2024
e7f4da0
Fix syllables doctest
jamesnohilly Nov 15, 2024
5cfffd0
Merge branch 'master' into jn/letters
jamesnohilly Nov 18, 2024
43532ba
Change documentation references
jamesnohilly Nov 20, 2024
a80667b
Add checks for PcpGroup
jamesnohilly Nov 25, 2024
30a5a62
Wrap used PcpGroup GAP functions
jamesnohilly Nov 25, 2024
2cd5cb9
Clean up docs examples
jamesnohilly Nov 25, 2024
5dd04b8
Clean and add tests for infinite PcGroups
jamesnohilly Nov 25, 2024
c07db31
Fix mistake in syllable test
jamesnohilly Nov 25, 2024
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54 changes: 54 additions & 0 deletions src/Groups/pcgroup.jl
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Expand Up @@ -365,3 +365,57 @@ function pc_group(c::GAP_Collector)
end
end

"""
letters(g::PcGroupElem)

Return the letters of `g` as a list of integers, each entry corresponding to
a group generator.
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@fingolfin fingolfin Nov 13, 2024 •
edited by lgoettgens
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Note that we can also produce negative numbers: e.g. -3 means "inverse of 3rd generator". This should be explained, and perhaps an example added showing that. E.g. based on this:

julia> x = (gg[1]*gg[2]*gg[3])^-2
g1*g2^-2*g3^3

Perhaps also add something like this (and then mirror it in the other function)

See also [`syllables`](@ref).

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I have added a small example with some brief explanation to letters for this. However I am unsure if the example is good as I was not able to get elements with negative exponents and test.

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For that you need an infinite group. E.g.

julia> g = dihedral_group(PosInf())
Pc group of infinite order

julia> g[1]^-3 * g[2]^-3
g1*g2^-3

or

julia> g = abelian_group(PcGroup, [5, 0])
Pc group of infinite order

julia> g[1]^-3 * g[2]^-3
g1^2*g2^-3


# Examples
```jldoctest
julia> c = collector(2, Int);

julia> Oscar.set_relative_orders!(c, [2, 3])

julia> Oscar.set_conjugate!(c, 2, 1, [2 => 2])

julia> gg = pc_group(c)
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Pc group of order 6

julia> letters(gg[1]^5*gg[2]^-4)
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3-element Vector{Int64}:
1
2
2
```
"""
function letters(g::PcGroupElem)
w = GAPWrap.UnderlyingElement(GapObj(g))
return Vector{Int}(GAPWrap.LetterRepAssocWord(w))
end

function Oscar.syllables(g::Union{PcGroupElem, SubPcGroupElem})
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l = GAPWrap.ExtRepOfObj(GapObj(g))
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For infinite groups, GAp uses a completely different internal representation. You'll need something like this:

  if GAP.Globals.IsPcpElement(GapObj(g))
     expvec = GAP.Globals.Exponents(GapObj(g))
     ... do something with it....
  else
    ... current code
  end

@assert iseven(length(l))
return Pair{Int, ZZRingElem}[l[i-1] => l[i] for i = 2:2:length(l)]
end

# Convert syllables in canonical form into exponent vector
#Thomas
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function exponent_vector(sylls::Vector{Pair{Int64, ZZRingElem}}, n)
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res = zeros(ZZRingElem, n)
for pair in sylls
@assert res[pair.first] == 0 #just to make sure
res[pair.first] = pair.second
end
return res
end

# Convert syllables in canonical form into group element
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#Thomas
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function (G::PcGroup)(sylls::Vector{Pair{Int64, ZZRingElem}})
e = exponent_vector(sylls, ngens(G))
pcgs = Oscar.GAPWrap.FamilyPcgs(GapObj(G))
x = Oscar.GAPWrap.PcElementByExponentsNC(pcgs, GapObj(e, true))
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Here you also need to check for if GAPWrap.IsPcGroup(GapObj(G)) and then do something different.

You'll need the function PcpElementByExponents or PcpElementByExponentsNC, see https://gap-packages.github.io/polycyclic/doc/chap4.html#X7882F0F57ABEB680

Also GAP.Globals.Collector(GapObj(G))

return Oscar.group_element(G, x)
end
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14 changes: 14 additions & 0 deletions test/Groups/pcgroup.jl
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Expand Up @@ -82,3 +82,17 @@ end
@test GAP.Globals.IsMutable(cgg)
@test cgg !== c.X
end

@testset "generate letters from polycyclic group element" begin

# finite polycyclic groups
c = collector(2, Int);
set_relative_order!(c, 1, 2)
set_relative_order!(c, 2, 3)
set_power!(c, 1, [2 => 1])
gg = pc_group(c)
@test letters(gg[1]^5*gg[2]^-4) == [1, 2, 2]
@test letters(gg[1]^5*gg[2]^4) == [1, 2] # all positive exp
@test letters(gg[1]^-5*gg[2]^-7) == [1, 2, 2] # all negative exp
@test letters(gg[1]^2*gg[2]^3) == [] # both identity elements
end
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