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Representation of the Alternating Group $A_4$: A4Irrep

A4Irrep represents irreducible representations (irreps) of the alternating group $A_4$ as a Sector. This page documents how the type behaves in code (construction, iteration, fusion, and access to topological data).

Sector type

The irreps of $A_4$ are labeled by integers 0:3. The first three labels are the one-dimensional representations $1$, $1'$, and $1''$; label 3 is the three-dimensional irrep $3$.

TensorKitSectors.A4Irrep — Type
struct A4Irrep <: AbstractIrrep{A₄}
A4Irrep(n::Integer)
Irrep[A₄](n::Integer)

Represents irreps of the alternating group $A₄$.

Fields

  • n::Int8: the label of the irrep, corresponding to $1$, $1′$, $1″$ and $3$.
source

The trivial sector is A4Irrep(0). Duals are

\[0^* = 0,\qquad 1^* = 2,\qquad 2^* = 1,\qquad 3^* = 3.\]

Fusion Rules

The one-dimensional irreps fuse as the group $\mathbb{Z}_3$:

\[a \otimes b = (a + b) \bmod 3,\qquad a,b \in \{0,1,2\}.\]

Fusing a one-dimensional irrep with the triplet gives the triplet again:

\[a \otimes 3 = 3 \otimes a = 3,\qquad a \in \{0,1,2\}.\]

The triplet product decomposes as

\[3 \otimes 3 = 0 \oplus 1 \oplus 2 \oplus 2 * 3.\]

Because the triplet appears with multiplicity two in $3 \otimes 3$, FusionStyle(A4Irrep) = GenericFusion(). The Nsymbol returns the corresponding integer multiplicity:

\[N_c^{ab} = \begin{cases} 1 + \delta_{c,3}, & a = b = 3,\\ \delta_{c,3}, & a = 3 \text{ or } b = 3,\\ \delta_{c,(a+b)\bmod 3}, & a,b \in \{0,1,2\}. \end{cases}\]

The quantum dimensions are the ordinary representation space dimensions:

\[d_0 = d_1 = d_2 = 1,\qquad d_3 = 3.\]

Topological Data

A4Irrep describes the fusion category $\mathsf{Rep(A_4)}$. The braiding is bosonic, and all topological twists are 1.

The Fsymbol is computed from the chosen fusiontensor basis, see below. The Rsymbol is the identity on all simple fusion channels, except for the two $3$ channels in $3 \otimes 3 \to 3$:

\[R^{33}_3 = \begin{pmatrix} -1 & 0\\ 0 & 1 \end{pmatrix}.\]

The first channel is the antisymmetric triplet and the second channel is the symmetric triplet.

Fusion Tensor and Basis Conventions

fusiontensor(a, b, c) returns a rank-4 array of size $d_a \times d_b \times d_c \times N_c^{ab}$. The one-dimensional fusion tensors are scalar phases/permutations in the selected basis. For $3 \otimes 3$, the singlet channels are normalized with a factor $1/\sqrt{3}$, and the two triplet channels use the real antisymmetric and symmetric Clebsch-Gordan tensors from the tetrahedral basis convention used in the implementation.

In this basis the triplet representation can be generated by

\[T_3 = \begin{pmatrix} 1 & 0 & 0\\ 0 & \omega & 0\\ 0 & 0 & \omega^2 \end{pmatrix}, \qquad S_3 = \frac{1}{3} \begin{pmatrix} -1 & 2 & 2\\ 2 & -1 & 2\\ 2 & 2 & -1 \end{pmatrix}, \qquad \omega = e^{2\pi i/3}.\]

The one-dimensional irreps have $S = 1$ and $T = \omega^n$ for label n. For every allowed fusion channel, the fusion tensor intertwines the product representation with the output representation.

using TensorKitSectors
using LinearAlgebra: kron
using Test: @test

omega = cis(2pi / 3)
T3 = [1 0 0; 0 omega 0; 0 0 omega^2]
S3 = 1 / 3 * [-1 2 2; 2 -1 2; 2 2 -1]

T(a::A4Irrep) = a.n == 3 ? T3 : hcat(omega^a.n)
S(a::A4Irrep) = a.n == 3 ? S3 : hcat(1)

a = A4Irrep(3)
b = A4Irrep(3)
c = A4Irrep(3)
C = fusiontensor(a, b, c)

for mu in 1:Nsymbol(a, b, c)
    Cmat = reshape(view(C, :, :, :, mu), dim(a) * dim(b), dim(c))
    @test Cmat' * kron(T(a), T(b)) * Cmat ≈ T(c)
    @test Cmat' * kron(S(a), S(b)) * Cmat ≈ S(c)
end

# output

References

For background on the group and the representation basis, see: