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https://github.com/kalmarek/SmallHyperbolic
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add logging and README.md
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[deps]
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AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
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Arpack = "7d9fca2a-8960-54d3-9f78-7d1dccf2cb97"
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Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
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DelimitedFiles = "8bb1440f-4735-579b-a4ab-409b98df4dab"
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GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
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Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"
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140
README.md
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140
README.md
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The repository contains code for running experiments for
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_Hyperbolic generalized triangle groups_ by
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[Pierre-Emmanuel Caprace](https://perso.uclouvain.be/pierre-emmanuel.caprace/),
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[Marston Conder](https://www.math.auckland.ac.nz/~conder/),
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[Marek Kaluba](https://kalmar.faculty.wmi.amu.edu.pl/) and
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[Stefan Witzel](https://www.math.uni-bielefeld.de/~switzel/).
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There are two disjoint computations covered in this repository.
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## Eigenvalues computations for _PSL₂(p)_
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This computations uses package
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[RamanujanGraphs.jl](https://github.com/kalmarek/RamanujanGraphs.jl) which
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implements (projective, special) linear groups of degree 2 (_PSL₂(p)_, _SL₂(p)_,
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_PGL₂(p)_ and _GL₂(p)_) and the irreducible representations for _SL₂(p)_.
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The script `adj_psl2_eigvals.jl` computes a subset of irreps of _SL₂(p)_ which
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descend to (mostly irreducible) representations of _PSL₂(p)_ in the following
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fashion.
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### Principal Series
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These representations are associated to the induced representations of _B(p)_,
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the _Borel subgroup_ (of upper triangular matrices) of _SL₂(p)_.
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All representations of the Borel subgroup come from the representations of the
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torus inside (i.e. diagonal matrices), hence are _1_-dimensional.
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Therefore to define a matrix representation of _SL₂(p)_ one needs to specify:
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* a complex character of 𝔽ₚ (finite field of _p_ elements)
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* an explicit set of representatives of _SL₂(p)/B(p)_.
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In code this can be specified by
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```julia
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p = 109 # our choice of a prime
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ζ = root_of_unity((p-1)÷2, ...) # ζ is (p-1)÷2 -th root of unity
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# two particular generators of SL₂(109):
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a = SL₂{p}([0 1; 108 11])
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b = SL₂{p}([57 2; 52 42])
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S = [a, b, inv(a), inv(b)] # symmetric generating set
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SL2p, _ = RamanujanGraphs.generate_balls(S, radius = 21)
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Borel_cosets = RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p}))
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# the generator of 𝔽ₚˣ
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α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
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ν₅ = let k = 5 # k runs from 0 to (p-1)÷4, or (p-3)÷4 depending on p (mod 4)
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νₖ = PrincipalRepr(
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α => ζ^k, # character sending α ↦ ζᵏ
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Borel_cosets
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)
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end
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```
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### Discrete Series
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These representations are associated with the action of _SL₂(p)_ (or in more
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generality of _GL₂(p)_) on ℂ[𝔽ₚ], the vector space of complex valued functions
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on 𝔽ₚˣ. There are however multiple choices how to encode such action.
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Let _L_ = 𝔽ₚ(√_α_) be the unique quadratic extension of 𝔽ₚ by a square of a
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generator _α_ of 𝔽ₚˣ. Comples characters of _Lˣ_ can be separated into
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_decomposable_ (the ones that take constant 1 value on the unique cyclic
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subgroup of order _(p+1)_ in _Lˣ_) and _nondecomposable_. Each _nondecomposable_
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character corresponds to a representation of _SL₂(p)_ in discrete series.
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To define matrix representatives one needs to specify
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* _χ_:𝔽ₚ⁺ → ℂ, a complex, non-trivial character of the _additive group_ of 𝔽ₚ
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* _ν_:_Lˣ_ → ℂ, a complex indecomposable character of _Lˣ_
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* a basis for ℂ[𝔽ₚ].
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Continuing the snippet above we can write
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```julia
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α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)) # a generator of 𝔽ₚˣ
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β = RamanujanGraphs.generator_min(QuadraticExt(α))
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# a generator of _Lˣ_ of minimal "Euclidean norm"
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ζₚ = root_of_unity(p, ...)
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ζ = root_of_unity(p+1, ...)
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ϱ₁₇ = let k = 17 # k runs from 1 to (p-1)÷4 or (p+1)÷4 depending on p (mod 4)
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DiscreteRepr(
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RamanujanGraphs.GF{p}(1) => ζₚ, # character of the additive group of 𝔽ₚ
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β => ζ^k, # character of the multiplicative group of _L_
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basis = [α^i for i in 1:p-1] # our choice for basis: the dual of
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)
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```
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A priori ζ needs to be a complex _(p²-1)_-th root of unity, however one can show
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that a reduction to _(p+1)_-th Cyclotomic field is possible.
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The script computing eigenvalues should be invoked by running
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```bash
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julia --project=. adj_psl2_eigvals.jl -p 109
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```
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The results will be written into `log` directory.
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## Sum of squares approach to property (T)
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> **NOTE**: This is mostly __unsuccessful computation__ as for none of the groups we examined
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the computations returned positive result (with the exception of Ronan's
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examples of groups acting on Ã₂-buildings).
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We try to find a sum of squares for various finitely presented groups using
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julia package [PropertyT.jl](https://github.com/kalmarek/PropertyT.jl). For
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full description of the method plesase refer to
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[1712.07167](https://arxiv.org/abs/1712.07167).
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The groups available are in the `./data` directory in files
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`presentations*.txt` files (in Magma format). For example
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```
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G_8_40_54_2 := Group< a, b, c |
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a^3, b^3, c^3,
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b*a*b*a,
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(c*b^-1*c*b)^2,
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(c^-1*b^-1*c*b^-1)^2,
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c*a*c^-1*a^-1*c^-1*a*c*a^-1,
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(c*a*c^-1*a)^3>
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```
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specifies group `G_8_40_54_2` as finitely presented group.
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The script needs GAP to be installed on the system (one can set `GAP_EXECUTABLE`
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environmental variable to point to `gap` exec). and tries to find both an
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automatic structure and a confluent Knuth-Bendix rewriting system on the given
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presentation. To attempt sum of squares method for proving property (T) one can
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execute
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```bash
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make 8_40_54_2
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```
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One can perform those computations in bulk by e.g. calling
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```bash
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make 2_4_4
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```
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to run all examples in `presentations_2_4_4.txt` in parallel.
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@ -2,21 +2,12 @@ using RamanujanGraphs
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using LinearAlgebra
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using Nemo
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using Logging
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using Dates
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include("src/nemo_utils.jl")
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const p = try
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@assert length(ARGS) == 2 && ARGS[1] == "-p"
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p = parse(Int, ARGS[2])
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RamanujanGraphs.Primes.isprime(p)
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p
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catch ex
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@error "You need to provide a prime `-p` which is congruent to 1 mod 4."
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rethrow(ex)
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end
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const CC = AcbField(256)
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SL2p = let
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function SL2p_gens(p)
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if p == 109
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a, b = let
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a = SL₂{p}([0 1; 108 11])
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@ -44,77 +35,99 @@ SL2p = let
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end
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end
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E, sizes =
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RamanujanGraphs.generate_balls([a, b, inv(a), inv(b)], radius = 21)
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@assert sizes[end] == RamanujanGraphs.order(SL₂{p})
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E
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return a,b
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end
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let Borel_cosets = Bcosets = RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p})),
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α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
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function adjacency(ϱ, CC, a, b)
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A = matrix(CC, ϱ(a))
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B = matrix(CC, ϱ(b))
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for j in 0:(p-1)÷4
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try
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h = PrincipalRepr(
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α => root_of_unity(CC, (p-1)÷2, j),
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Borel_cosets)
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return sum(A^i for i = 1:4) + sum(B^i for i = 1:4)
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end
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@time adjacency = let
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A = matrix(CC, h(SL2p[2]))
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B = matrix(CC, h(SL2p[3]))
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sum(A^i for i in 1:4) + sum(B^i for i in 1:4)
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end
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const p = try
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@assert length(ARGS) == 2 && ARGS[1] == "-p"
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p = parse(Int, ARGS[2])
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RamanujanGraphs.Primes.isprime(p)
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p
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catch ex
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@error "You need to provide a prime `-p` which is congruent to 1 mod 4."
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rethrow(ex)
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end
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@time ev = let evs = safe_eigvals(adjacency)
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_count_multiplicites(evs)
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end
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if length(ev) == 1
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@info "Principal Series Representation $j" ev[1]
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else
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@info "Principal Series Representation $j" ev[1:2] ev[end]
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end
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catch ex
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@error "Principal Series Representation $j failed" ex
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ex isa InterruptException && rethrow(ex)
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const LOGFILE = "SL(2,$p)_eigvals_$(now()).log"
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open(joinpath("log", LOGFILE), "w") do io
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with_logger(SimpleLogger(io)) do
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CC = AcbField(128)
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a,b = SL2p_gens(p)
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Borel_cosets = let p = p, (a,b) = (a,b)
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SL2p, sizes =
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RamanujanGraphs.generate_balls([a, b, inv(a), inv(b)], radius = 21)
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@assert sizes[end] == RamanujanGraphs.order(SL₂{p})
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RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p}))
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end
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end
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end
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let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)),
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β = RamanujanGraphs.generator_min(QuadraticExt(α))
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let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
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if p % 4 == 1
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ub = (p - 1) ÷ 4
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ζ = root_of_unity(CC, (p + 1) ÷ 2, (p - 1) ÷ 4)
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else # p % 4 == 3
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ub = (p + 1) ÷ 4
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ζ = root_of_unity(CC, (p + 1), 1)
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end
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for j = 0:(p-1)÷4
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h = PrincipalRepr(
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α => root_of_unity(CC, (p - 1) ÷ 2, j),
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Borel_cosets,
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)
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for k = 1:ub
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try
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h = DiscreteRepr(
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RamanujanGraphs.GF{p}(1) => root_of_unity(CC, p),
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β => ζ^k,
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)
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@time adj = adjacency(h, CC, a, b)
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@time adjacency = let
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A = matrix(CC, h(SL2p[2]))
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B = matrix(CC, h(SL2p[3]))
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sum(A^i for i = 1:4) + sum(B^i for i = 1:4)
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try
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@time ev = let evs = safe_eigvals(adj)
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_count_multiplicites(evs)
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end
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@info "Principal Series Representation $j" ev[1:2] ev[end]
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catch ex
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@error "Principal Series Representation $j failed" ex
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ex isa InterruptException && rethrow(ex)
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end
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end
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@time ev = let evs = safe_eigvals(adjacency)
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_count_multiplicites(evs)
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end
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@info "Discrete Series Representation $k" ev[1:2] ev[end]
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catch ex
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@error "Discrete Series Representation $k : failed" ex
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ex isa InterruptException && rethrow(ex)
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end
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end
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end
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let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)),
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β = RamanujanGraphs.generator_min(QuadraticExt(α))
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if p % 4 == 1
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ub = (p - 1) ÷ 4
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ζ = root_of_unity(CC, (p + 1) ÷ 2, 1)
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else # p % 4 == 3
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ub = (p + 1) ÷ 4
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ζ = root_of_unity(CC, (p + 1), 1)
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end
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for k = 1:ub
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h = DiscreteRepr(
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RamanujanGraphs.GF{p}(1) => root_of_unity(CC, p),
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β => ζ^k,
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)
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@time adj = adjacency(h, CC, a, b)
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try
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@time ev = let evs = safe_eigvals(adj)
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_count_multiplicites(evs)
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end
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@info "Discrete Series Representation $k" ev[1:2] ev[end]
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catch ex
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@error "Discrete Series Representation $k : failed" ex
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ex isa InterruptException && rethrow(ex)
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end
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end
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end
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end # with_logger
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end # open(logfile)
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#
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# using RamanujanGraphs.LightGraphs
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