Following paper

Ramanujan graphs by Lubotzky, A., Phillips, R. & Sarnak, P. Combinatorica (1988) 8: 261. https://doi.org/10.1007/BF02126799

this package implements function lps(p::Integer, q::Integer) for different primes p,q congruent to 1 modulo 4 returning the appropriate Cayley graph.

A basic syntax is as follows:

using RamanujanGraphs
G, verts, vlabels, elabels = lps(p, q)

where

• G is (p+1)-regular graph with
  • q³ - q vertices if p is not a square modulo q (Cayley graph of PGL₂(q))
  • (q³ - q)/2 vertices if p is a square modulo q (Cayley graph of PSL₂(q))
• verts is a plain array of vertices (=group elements)
• vlabels is a labelling dictionary for vertices: group element pointing to its vertex in the graph
• elabels is a dictionary for edges:
  • a tuple of integers (src, dst) points to a generator g of the group iff verts[src]^-1*verts[dst] == g, i.e. if we travel from src to dst by multiplying src by g on the right. Note that only one of (src, dst) and (dst, src) is stored.

Timings:

julia> using RamanujanGraphs, RamanujanGraphs.Graphs

julia> let (p,q) = (13,61)
           lps(p, q);
           @time G, verts, vlabels, elabels = lps(p, q);
           @assert nv(G) == RamanujanGraphs.order(eltype(verts))
           @info "Cayley graph of $(eltype(verts)):" degree=length(neighbors(G,1)) size=nv(G)
       end
 1.701829 seconds (2.05 M allocations: 272.833 MiB)
┌ Info: Cayley Graph of PSL₂{61}:
│   degree = 14
└   size = 113460

julia> let (p,q) = (13,73)
           lps(p, q);
           @time G, verts, vlabels, elabels = lps(p, q);
           @assert nv(G) == RamanujanGraphs.order(eltype(verts))
           @info "Cayley graph of $(eltype(verts)):" degree=length(neighbors(G,1)) size=nv(G)
       end
 6.400727 seconds (6.68 M allocations: 655.549 MiB)
┌ Info: Cayley Graph of PGL₂{73}:
│   degree = 14
└   size = 388944

Irreducible representations for SL₂(p)

Principal Series

These representations are associated to the induced representations of B(p), the Borel subgroup (of upper triangular matrices) of SL₂(p). All representations of the Borel subgroup come from the representations of the torus inside (i.e. diagonal matrices), hence are 1-dimensional.

Therefore to define a matrix representation of SL₂(p) one needs to specify:

• a complex character of 𝔽ₚ (finite field of p elements)
• an explicit set of representatives of SL₂(p)/B(p).

In code this can be specified by

p = 109 # our choice of a prime
ζ = root_of_unity((p-1)÷2, ...) # ζ is (p-1)÷2 -th root of unity
# two particular generators of SL₂(109):
a = SL₂{p}([0 1; 108 11])
b = SL₂{p}([57 2; 52 42])

S = [a, b, inv(a), inv(b)] # symmetric generating set
SL2p, _ = RamanujanGraphs.generate_balls(S, radius = 21)

Borel_cosets = RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p}))
# the generator of 𝔽ₚˣ
α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))

ν₅ = let k = 5 # k runs from 0 to (p-1)÷4, or (p-3)÷4 depending on p (mod 4)
  νₖ = PrincipalRepr(
      α => ζ^k, # character sending α ↦ ζᵏ
      Borel_cosets
    )
end

Discrete Series

These representations are associated with the action of SL₂(p) (or in more generality of GL₂(p)) on ℂ[𝔽ₚˣ], the vector space of complex valued functions on 𝔽ₚˣ. There are however multiple choices how to encode such action.

Let L = 𝔽ₚ(√_α_) be the unique quadratic extension of 𝔽ₚ by a square of a generator α of 𝔽ₚˣ. Comples characters of Lˣ can be separated into decomposable (the ones that take constant 1 value on the unique cyclic subgroup of order (p+1) in Lˣ) and nondecomposable. Each nondecomposable character corresponds to a representation of SL₂(p) in discrete series.

To define matrix representatives one needs to specify

• χ:𝔽ₚ⁺ → ℂ, a complex, non-trivial character of the additive group of 𝔽ₚ
• ν:Lˣ → ℂ, a complex indecomposable character of Lˣ
• a basis for ℂ[𝔽ₚ].

Continuing the snippet above we can write

α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)) # a generator of 𝔽ₚˣ
β = RamanujanGraphs.generator_min(QuadraticExt(α))
# a generator of _Lˣ_ of minimal "Euclidean norm"

ζₚ = root_of_unity(p, ...)
ζ = root_of_unity(p+1, ...)

ϱ₁₇ = let k = 17 # k runs from 1 to (p-1)÷4 or (p+1)÷4 depending on p (mod 4)
    DiscreteRepr(
    RamanujanGraphs.GF{p}(1) => ζₚ, # character of the additive group of 𝔽ₚ
    β => ζ^k, # character of the multiplicative group of _L_
    basis = [α^i for i in 1:p-1] # our choice for basis: the dual of
)

A priori ζ needs to be a complex (p²-1)-th root of unity, however one can show that a reduction to (p+1)-th Cyclotomic field is possible.