Representation

A (1D) irreducible representation of a finite group. Upon creation, the group homomorphism properties are verified.

Sources
representation.hpp
representation.cpp
representation.jl

Constructors

Trivial representation

Creates the trivial representation (all characters equal to 1) of a PermutationGroup

C++Julia

Representation(PermutationGroup const &group);

Representation(group::PermutationGroup)

With characters

Creates a 1D representation of a PermutationGroup with given real or complex characters.

C++Julia

template <typename T>
Representation(PermutationGroup const &group, std::vector<T> const &characters);
template <typename T>
Representation(PermutationGroup const &group, arma::Col<T> const &characters);
template <typename T>
Representation(PermutationGroup const &group, T *characters, int64_t n_characters);

Representation(group::PermutationGroup, characters::Vector{Float64})
Representation(group::PermutationGroup, characters::Vector{ComplexF64})

Name	Description
group	PermutationGroup of the Representation
characters	characters of the representation
n_characters	length of the array of characters

The template parameter T in C++ can either be double or complex.

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Methods

size

Returns the size of the Representation, i.e. the number of group elements represented.

C++Julia

int64_t size(Representation const &irrep);

size(irrep::Representation)::Int64

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isreal

Returns the whether or not the Representation is real, I.E. the characters are real numbers and do not have an imaginary part.

C++Julia

bool isreal(Representation const &irrep) const;

isreal(irrep::Representation)::Bool

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* operator

Multiplies two Representations by overloading the * operator.

C++Julia

Representation operator*(Representation const &r1, Representation const &r2);

Base.:*(r1::Representation, r2::Representation)::Representation

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to_string (operator<<)

Converts the Representation to a readable string representation.

C++Julia

std::string to_string(Representation const &irrep);
std::ostream &operator<<(std::ostream &out, Representation const &irrep);

to_string(irrep::Representation)::String

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Usage Example

C++Julia

Permutation p = {1, 2, 3, 0};
auto C4 = PermutationGroup({pow(p, 0), pow(p, 1), pow(p, 2), pow(p, 3)});
Representation r1(C4, arma::vec{1.0, -1.0, 1.0, -1.0});
Representation r2(C4, arma::cx_vec{1.0, 1.0i, -1.0, -1.0i});
XDIAG_SHOW(r1 * r2);

p = Permutation([2, 3, 4, 1])
C4 = PermutationGroup([p^0, p^1, p^2, p^3])
r1 = Representation(C4, [1.0, -1.0, 1.0, -1.0])
r2 = Representation(C4, [1.0, 1.0im, -1.0, -1.0im])
@show r1 * r2