A Julia package for representation theory of the symmetric group
This package supports basic representation theory of the symmetric group. One can form irreducible representations (irreps) by specifying the corresponding permutation, combine representations via direct sum and Kronecker product, and also calculate the resulting irrep multipliciplities. For example, the following code calculates the Kronecker coefficients of two irreps of S₇, specified by the partitions 5+1+1 and 2+2+2+1:
julia> using NumericalRepresentationTheory, Permutations, Plots
julia> R₁ = Representation(5,1,1);
julia> g = Permutation([7,2,1,3,4,6,5]); Matrix(R₁(g)) # Matrix representation of a specific permutation
15×15 Array{Float64,2}:
-0.2 -0.0408248 -0.0527046 … 0.0 0.0 0.0
0.163299 0.241667 0.31199 0.0 0.0 0.0
0.0 -0.161374 0.0138889 0.0 0.0 0.0
0.0 0.0 -0.157135 0.467707 0.810093 0.0
0.0 0.0 0.0 0.270031 -0.155902 -0.881917
-0.966092 0.0493007 0.0636469 … 0.0 0.0 0.0
0.0 -0.190941 0.0164336 0.0 0.0 0.0
0.0 0.0 -0.185924 0.0790569 0.136931 0.0
0.0 0.0 0.0 0.0456435 -0.0263523 -0.149071
0.0 0.935414 -0.0805076 0.0 0.0 0.0
0.0 0.0 0.91084 … 0.0968246 0.167705 0.0
0.0 0.0 0.0 0.0559017 -0.0322749 -0.182574
0.0 0.0 0.0 0.125 0.216506 0.0
0.0 0.0 0.0 0.0721688 -0.0416667 -0.235702
0.0 0.0 0.0 -0.816497 0.471405 -0.333333
julia> R₂ = Representation(2,2,2,1);
julia> R = R₁ ⊗ R₂; # Tensor product representation
julia> multiplicities(R) # Returns a dictionary whose keys are partitions and values are the multiplicities
Dict{Partition, Int64} with 8 entries:
7 = 2 + 2 + 2 + 1 => 1
7 = 4 + 2 + 1 => 1
7 = 3 + 1 + 1 + 1 + 1 => 1
7 = 3 + 2 + 2 => 1
7 = 3 + 3 + 1 => 1
7 = 2 + 2 + 1 + 1 + 1 => 1
7 = 4 + 1 + 1 + 1 => 1
7 = 3 + 2 + 1 + 1 => 2
julia> plot(multiplicities(R)) # We can also plot
In addition, one can find an orthogonal transformation that reduces a representation to irreducibles:
julia> λ,Q = blockdiagonalize(R); # Q'R(g)*Q ≈ ρ(g) where ρ = Representation(λ) is a direct sum (block diagonal) of irreducibles encoded by the dictionary λ.
julia> ρ = Representation(λ);
julia> Q'R(g)*Q ≈ ρ(g)
true
julia> ρ(g) ≈ (Representation(4,2,1) ⊕ Representation(4,1,1,1) ⊕ Representation(3,3,1) ⊕ Representation(3,2,2) ⊕ Representation(3,2,1,1) ⊕ Representation(3,2,1,1) ⊕ Representation(3,1,1,1,1) ⊕ Representation(2,2,2,1) ⊕ Representation(2,2,1,1,1))(g)
true