Representation of the finite Heisenberg Group $H_N$: HeisenbergIrrep{N}

HeisenbergIrrep{N} represents irreducible representations (irreps) of the finite Heisenberg group $H_N$ as a Sector. $H_N$ is the group of $3× 3$ upper triangular matrices which is identified with the group $ℤ_N ⋊ ℤ_N^2$; it is only defined here for prime N. This page documents how the type behaves in code (construction, iteration, fusion, and access to topological data).

Sector type

struct HeisenbergIrrep{N} <: AbstractIrrep{Heisenberg{N}}
HeisenbergIrrep{N}(n::Int8)
Irrep[Heisenberg{N}](n::Int8)

Every irrep is labeled by a single integer n::Int8, but the meaning of n depends on its sign:

• If 0 <= n < N^2, the irrep is the one-dimensional character $χ_{a,b}$ with a, b = divrem(n, N). There are $N^2$ such characters, one for each pair (a, b) ∈ 0:N-1.
• If -N < n < 0, the irrep is the $N$-dimensional Schrödinger representation $π_k$ with k = -n, k ∈ 1:N-1.

In total there are $N^2 + N - 1$ irreps (the trivial $k = 0$ Schrödinger representation is excluded, since it is reducible and decomposes into the $N^2$ characters). The alias Heis3Irrep = HeisenbergIrrep{3} is provided for the smallest (and most commonly used) case.

The trivial sector is HeisenbergIrrep{N}(0), i.e. $χ_{0,0}$. Duals are

\[χ_{a,b}^* = χ_{-a \bmod N,\,-b \bmod N}, \qquad π_k^* = π_{N-k}.\]

Fusion Rules

Characters fuse among themselves as the abelian group $ℤ_N × ℤ_N$:

\[χ_{a,b} ⊗ χ_{a',b'} = χ_{(a+a')\bmod N,\,(b+b')\bmod N}.\]

Fusing a character with a Schrödinger irrep leaves the latter unchanged, regardless of the character's labels:

\[χ_{a,b} ⊗ π_k = π_k ⊗ χ_{a,b} = π_k.\]

This reflects that $π_k$ is the unique irrep with a given nontrivial central character, so twisting it by any character of the full group produces an isomorphic representation.

Two Schrödinger irreps fuse according to whether their labels cancel modulo N:

\[π_k ⊗ π_{k'} = \begin{cases} N ⋅ π_{(k+k') \bmod N}, & k + k' ≢ 0 \pmod N, \\[4pt] \displaystyle\bigoplus_{a,b=0}^{N-1} χ_{a,b}, & k + k' ≡ 0 \pmod N. \end{cases}\]

In the first case the $N$-fold multiplicity accounts for the dimension count $N × N = N^2$ ($N$ copies of the $N$-dimensional $π_{(k+k')\bmod N}$); in the second, the $N^2$-dimensional product decomposes into all $N^2$ one-dimensional characters, each with multiplicity one.

Because of this multiplicity-$N$ channel, FusionStyle(HeisenbergIrrep) = GenericFusion(). The Nsymbol returns the corresponding integer multiplicities directly from the case distinction above.

Quantum dimensions are the ordinary representation space dimensions:

\[d_{χ_{a,b}} = 1, \qquad d_{π_k} = N.\]

Topological Data

BraidingStyle(HeisenbergIrrep) = Bosonic(): braiding is symmetric, and braidingscalartype(HeisenbergIrrep) = ComplexF64 since the Clebsch-Gordan coefficients are generally complex.

The Fsymbol and Rsymbol are both computed generically from the chosen fusiontensor basis (via Fsymbol_from_fusiontensor and Rsymbol_from_fusiontensor), see below. On the single-channel sectors ($χ ⊗ χ → χ$ and $χ ⊗ π → π$) the R-symbol evaluates to the trivial phase 1. On the multiplicity-$N$ channel $π_k ⊗ π_{k'} → π_{(k+k')\bmod N}$ and on the channels $π_k ⊗ π_{-k} → χ_{a,b}$, the R-symbol carries nontrivial phases built out of N-th roots of unity.

Fusion Tensor and Basis Conventions

fusiontensor(a, b, c) returns a rank-4 array of size $d_a × d_b × d_c × N_c^{ab}$. The one-dimensional (character) fusion tensors are trivial scalars. For the Schrödinger irreps, the fusion tensor is built out of the $N$-th root of unity $ω = e^{2π i/N}$, following the reference below:

• $π_k ⊗ π_{k'} → π_{(k+k')\bmod N}$ ($k+k' ≢ 0$): a permutation tensor (with a phase for the case where one of the factors is a character, see below).
• $π_k ⊗ π_{-k} → χ_{a,b}$: normalized with a factor $1/\sqrt{N}$, with phase $ω^{-a i}$ along the surviving diagonal.
• $χ_{a,b} ⊗ π_k → π_k$ (and the symmetric $π_k ⊗ χ_{a,b} → π_k$): a permutation tensor dressed with the phase $ω^{a m}$.

The representation can be checked against the generators X and Z of $H_N$ acting as $X|i\rangle = |i+1\rangle$ and $Z|i\rangle = ω^{ki}|i\rangle$ on $π_k$, and as $X ↦ ω^a$, $Z ↦ ω^b$ on $χ_{a,b}$. For every allowed fusion channel, the fusion tensor acts as an intertwiner between the tensor product representation and the fused representation:

using TensorKitSectors
using LinearAlgebra: kron, Diagonal
using Test: @test

N = 3
ω = cispi(2 / N)

function X(s::HeisenbergIrrep{N}) where {N}
    s.n < 0 || return hcat(ω^div(s.n, N))
    M = zeros(ComplexF64, N, N)
    for i in 0:(N - 1)
        M[mod(i + 1, N) + 1, i + 1] = 1
    end
    return M
end
Z(s::HeisenbergIrrep{N}) where {N} = s.n < 0 ? Diagonal([ω^(-s.n * i) for i in 0:(N - 1)]) : hcat(ω^mod(s.n, N))

for a in values(HeisenbergIrrep{N}), b in values(HeisenbergIrrep{N})
    for c in a ⊗ b
        C = fusiontensor(a, b, c)
        for μ in 1:Nsymbol(a, b, c)
            Cmat = reshape(view(C, :, :, :, μ), dim(a) * dim(b), dim(c))
            @test Cmat' * kron(X(b), X(a)) * Cmat ≈ X(c)
            @test Cmat' * kron(Z(b), Z(a)) * Cmat ≈ Z(c)
        end
    end
end

# output

Iteration

Iterating over values(HeisenbergIrrep{N}) first yields the $N^2$ characters $χ_{0,0}, χ_{0,1}, …$ (in order of increasing n), followed by the Schrödinger irreps $π_1, …, π_{N-1}$.

References

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

• Heisenberg group
• Bacon, D., How a Clebsch-Gordan Transform Helps to Solve the Heisenberg Hidden Subgroup Problem