Introduction

DecomposingGroupRepresentations.jl is a Julia package that provides an API for decomposing representations of reductive groups acting on multivariate polynomials using DynamicPolynomials.jl.

Quick start

julia> using DecomposingGroupRepresentations
julia> @polyvar x y z(x, y, z)
julia> vars = [x, y, z]3-element Vector{Variable{Commutative{CreationOrder}, Graded{LexOrder}}}:
 x
 y
 z
julia> SO3 = LieGroup("SO", 3)LieGroup SO(3)
 number type (or field): ComplexF64
 weight type: Int64
 Lie algebra properties:
  dimension: 3
  rank (dimension of Cartan subalgebra): 1
julia> a = MatrixGroupAction(SO3, [vars])MatrixGroupAction of SO(3)
 1 vectors under action: [x, y, z]
julia> V = FixedDegreePolynomials(vars, 2)FixedDegreePolynomials space of dimension 6
 variables: x, y, z
 degree: 2
julia> ρ = GroupRepresentation(a, V)GroupRepresentation of SO(3) on 6-dimensional vector space
 Lie group: SO(3)
julia> irrs = IrreducibleDecomposition(ρ)IrreducibleDecomposition of SO(3)-action on 6-dimensional vector space
 number of irreducibles: 2
 dimensions of irreducibles: 1, 5
julia> [highest_weight(irr) for irr in irreducibles(irrs)]2-element Vector{Weight{Int64}}:
 Weight: [0]
 Weight: [2]
julia> [vector(hw_vector(irr)) for irr in irreducibles(irrs)]2-element Vector{Polynomial{Commutative{CreationOrder}, Graded{LexOrder}, ComplexF64}}:
 z² + y² + x²
 -y² + (0.0 + 2.0im)xy + x²
julia> iso = IsotypicDecomposition(ρ)IsotypicDecomposition of SO(3)-action on 6-dimensional vector space
 number of isotypic components: 2
 multiplicities of irreducibles: 1, 1
 dimensions of irreducibles: 1, 5
 dimensions of isotypic components: 1, 5
julia> basis(iso[Weight([0])]) # invariant1-element Vector{Polynomial{Commutative{CreationOrder}, Graded{LexOrder}, ComplexF64}}:
 z² + y² + x²
julia> H₂ = basis(iso[Weight([2])]) # spherical harmonics5-element Vector{Polynomial{Commutative{CreationOrder}, Graded{LexOrder}, ComplexF64}}:
 -y² + (0.0 + 2.0im)xy + x²
 (0.0 + 1.0im)yz + xz
 (2.0 + 0.0im)z² + -y² + -x²
 (0.0 + 1.0im)yz + -xz
 -y² + (-0.0 - 2.0im)xy + x²
julia> rref(H₂)Computing rref, merging monomials...
Monomials merged
Constructing coefficients matrix...
Coefficients matrix constructed
RREF computed
5-element Vector{Polynomial{Commutative{CreationOrder}, Graded{LexOrder}, ComplexF64}}:
 z² + -x²
 yz
 y² + -x²
 xz
 xy