Dihedral Group Representations: DNIrrep
DNIrrep{N} represents irreducible representations of the dihedral group $D_N = \mathbb{Z}_N ⋊ C$. The aliases D3Irrep and D4Irrep are provided for DNIrrep{3} and DNIrrep{4}.
Sector type
TensorKitSectors.DNIrrep — Type
struct DNIrrep{N} <: AbstractIrrep{Dihedral{N}}
DNIrrep{N}(n::Integer, isodd::Bool=false)
Irrep[Dihedral{N}](n::Integer, isodd::Bool=false)Represents irreps of the dihedral group $D_N = Z_N ⋊ C$ ($Z_N$ and charge conjugation or reflection).
Properties
j::Int: the value of the $Z_N$ charge.isodd::Bool: the representation of charge conjugation.
Combined these take the values $+0, -0, 1, ..., (N - 1) / 2$ for odd $N$, and $+0, -0, 1, ..., N / 2 - 1, +(N/2), -(N/2)$ for even $N$, where the $+$ ($-$) refer to the even (odd) one-dimensional irreps, while the others are two-dimensional.
Labels are constructed as DNIrrep{N}(j, isodd=false). The integer j satisfies 0 <= j <= N ÷ 2. The isodd flag is valid only for one-dimensional irreps: always at j == 0, and also at j == N/2 when N is even.
For odd N, the labels are
\[(0,\text{false}),\ (0,\text{true}),\ 1,2,\ldots,(\frac{N-1}{2},\text{false}).\]
For even N, the labels are
\[(0,\text{false}),\ (0,\text{true}),\ 1,2,\ldots,\frac{N}{2}-1,\ (N/2,\text{false}),\ (N/2,\text{true}).\]
The unit is DNIrrep{N}(0, false), and all irreps are self-dual.
Fusion Rules
The one-dimensional irreps fuse by XOR of the parity label. The fusion product of a one-dimensional irrep with a two-dimensional irrep preserves the two-dimensional label, up to the special one-dimensional labels at j == N/2 for even N.
For ordinary two-dimensional labels, fusion follows the dihedral character rule:
\[\rho_i \otimes \rho_j = \rho_{|i-j|} \oplus \rho_{i+j},\]
where labels are folded back into the range 0:N÷2. When a folded output lands on a one-dimensional point (0, or N/2 for even N), it splits into the corresponding even and odd one-dimensional irreps.
For N < 3, FusionStyle(DNIrrep{N}) = UniqueFusion(). Otherwise, FusionStyle(DNIrrep{N}) = SimpleFusion().
Quantum dimensions are
\[d_{(0,\pm)} = 1,\qquad d_{\rho_j} = 2,\qquad d_{(N/2,\pm)} = 1\quad (N \text{ even}).\]
Topological Data
DNIrrep{N} is a bosonic representation category, so all twists are 1. The Fsymbol is computed from the selected fusiontensor basis. The Rsymbol is the N-symbol with a possible sign on odd one-dimensional output channels from two non-trivial two-dimensional inputs.
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 implementation uses real Clebsch-Gordan coefficients. #TODO: source for the convention used here
Iteration
The DNIrrep{N} labels are iterable, with the same ordering as described above.