The hypercubic group

Theory, part 1 of 3. Next: Clebsch-Gordan coefficients · Lattice operators

This page introduces the hypercubic group ${H(4)}$, its irreducible representations and how they combine in tensor products. Throughout the wiki, the irrep written ${\tau^{(k)}_l}$ in the literature is the label (k, l) in the code: for example, ${\tau^{(4)}_1}$ is (4, 1).

Elements of H(4)

In four-dimensional Euclidean space, the group of rotations and reflections is the continuous group ${O(4)}$. On a hypercubic lattice only the elements of ${O(4)}$ that map the lattice onto itself survive. They form a finite subgroup, the hypercubic group ${H(4)}$.

Each of these elements maps every axis onto plus or minus another axis: it is a signed permutation of the four axes, i.e. a permutation of the axis labels combined with a change of sign of some of the axes. The permutations form the symmetric group ${S_4}$, of order ${4! = 24}$. The sign changes form the group ${(\mathbb{Z}_2)^4}$, of order ${2^4 = 16}$, where ${\mathbb{Z}_2}$ is the cyclic group of order two. Conjugating a sign change with a permutation gives another sign change, so the sign changes form a normal subgroup and ${H(4)}$ is the semidirect product

$$H(4) = (\mathbb{Z}_2)^4 \rtimes S_4 , \qquad |H(4)| = 2^4 \cdot 4! = 384 .$$

Not all the elements are rotations: the swap of two axes and the reflection of one axis have determinant ${-1}$. The rotations are the 192 elements with determinant ${+1}$.

Generators

All the elements of ${H(4)}$ can be generated from three of them, ${\alpha}$, ${\beta}$ and ${\gamma}$. We adopt the notation of Baake et al. (1982) and Göckeler et al. (1996):

• ${\alpha}$ swaps the first two axes: it is the transposition ${(12)}$;
• ${\beta}$ permutes the four axes cyclically, ${1 \to 2 \to 3 \to 4 \to 1}$: it is the 4-cycle ${(1234)}$;
• ${\gamma}$ reflects the first axis, ${x_1 \to -x_1}$.

The first two generate the permutations ${S_4}$ and satisfy its defining relations ${\alpha^2 = \beta^4 = (\alpha\beta)^3 = 1}$ (the product ${\alpha\beta}$ is a 3-cycle). The last one, ${\gamma}$, generates the ${\mathbb{Z}_2}$ structure: combined with the permutations, it gives the reflection of every axis. In the vector representation ${\tau^{(4)}_1}$, where the group acts on the four axes, the generators are the matrices

$$\alpha = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad \beta = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}, \qquad \gamma = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} .$$

For each of the 20 irreps of ${H(4)}$ (see below), Baake et al. (1982) give explicit matrices for the three generators.

Note. In Table IXB of Baake et al. (1982), the matrix given for ${\alpha}$ in the irrep ${\tau^{(2)}_2}$ is the identity. This cannot be right: the matrices of every representation must satisfy ${(\alpha\beta)^3 = 1}$, because ${\alpha\beta}$ is a 3-cycle of ${S_4}$, while with ${\alpha = 1}$ one gets ${(\alpha\beta)^3 = \beta^3 \neq 1}$. Moreover ${\tau^{(2)}_2 = \tau^{(2)}_1 \otimes \tau^{(1)}_2}$, and ${\tau^{(1)}_2}$ is trivial on the permutations, so ${\alpha}$ must be the same as in ${\tau^{(2)}_1}$: h4lat uses ${\alpha = \mathrm{diag}(1, -1)}$. (Until October 2026, h4lat used the identity, as in the reference.)

The 384 matrices

From these ${20 \cdot 3}$ generator matrices, h4lat builds the explicit matrices of all the elements in every irrep, ${20 \cdot 384}$ matrices in total. Starting from the identity, ${\alpha}$ and ${\beta}$ are applied iteratively until the ${4! = 24}$ distinct elements of ${S_4}$ have been generated (new elements are recognised by their matrix in ${\tau^{(4)}_1}$, and the same products are formed in every irrep). Looping over these 24 elements ${g}$, we obtain all the remaining ones as the products

$$g \left(\gamma^{(1)}\right)^{i_1} \left(\gamma^{(2)}\right)^{i_2} \left(\gamma^{(3)}\right)^{i_3} \left(\gamma^{(4)}\right)^{i_4}, \qquad i_1, i_2, i_3, i_4 \in \{0, 1\}, \tag{1}$$

where ${\gamma^{(j)}}$ reflects the ${j}$-th axis. It is obtained from ${\gamma}$ by conjugation with ${\beta}$:

$$\gamma^{(1)} \equiv \gamma, \qquad \gamma^{(j)} \equiv \beta^{\,j-1} \, \gamma \, \beta^{-(j-1)} \quad (j = 2, 3, 4):$$

${\beta^{-(j-1)}}$ maps the ${j}$-th axis onto the first one, ${\gamma}$ reflects it, and ${\beta^{j-1}}$ maps it back. In this way, ${24 \cdot 2^4 = 384}$ elements are generated for each irrep.

In the code. The generator matrices are the lists alpha_list, beta_list and gamma_list of the module h4lat.cg_calculator, in the order of rep_label_list. The matrices of all the elements are shipped with the package (folder src/h4lat/data/h4_ele/) and are held by the attribute h4_mat of any cg_calc object: h4_mat[ir][e] is the matrix of element e in the irrep rep_label_list[ir], where e = 16*s + 8*i1 + 4*i2 + 2*i3 + i4 for the s-th element of ${S_4}$ and the exponents of Eq. (1).

from h4lat import cg_calc, irrep_index
from h4lat.cg_calculator import alpha_list, beta_list, gamma_list

h4_mat = cg_calc((4, 1), (4, 1), verbose=False).h4_mat  # a bundled product: nothing is computed
vector = h4_mat[irrep_index[(4, 1)]]                    # the 384 matrices of the vector irrep (4,1)
print(vector[8])                                        # γ⁽¹⁾ = diag(-1, 1, 1, 1)
print(alpha_list[irrep_index[(2, 2)]])                  # α in (2,2): diag(1, -1)

Irreducible representations

${H(4)}$ has 20 conjugacy classes, and therefore 20 irreducible representations (irreps). Writing ${\tau^{(k)}_l}$ for the ${l}$-th irrep of dimension ${k}$, they are

$$\tau^{(1)}_1,\ \tau^{(1)}_2,\ \tau^{(1)}_3,\ \tau^{(1)}_4,\quad \tau^{(2)}_1,\ \tau^{(2)}_2,\quad \tau^{(3)}_1,\ \tau^{(3)}_2,\ \tau^{(3)}_3,\ \tau^{(3)}_4,\quad \tau^{(4)}_1,\ \tau^{(4)}_2,\ \tau^{(4)}_3,\ \tau^{(4)}_4,\quad \tau^{(6)}_1,\ \tau^{(6)}_2,\ \tau^{(6)}_3,\ \tau^{(6)}_4,\quad \tau^{(8)}_1,\ \tau^{(8)}_2 .$$

As for every finite group, the squares of the dimensions add up to the order of the group:

$$4 \cdot 1^2 + 2 \cdot 2^2 + 4 \cdot 3^2 + 4 \cdot 4^2 + 4 \cdot 6^2 + 2 \cdot 8^2 = 384 .$$

The irreps of the five Dirac structures of the lattice operators have a direct physical meaning:

Irrep	Code label	Dim.	What transforms in it
${\tau^{(1)}_1}$	(1, 1)	1	scalars (the trivial irrep), e.g. ${\bar\psi\psi}$
${\tau^{(1)}_4}$	(1, 4)	1	pseudoscalars (the determinant of the signed permutation), e.g. ${\bar\psi\gamma_5\psi}$
${\tau^{(4)}_1}$	(4, 1)	4	vectors (the action on the four axes), e.g. ${\gamma_\mu}$ and the derivative ${D_\mu}$
${\tau^{(4)}_4}$	(4, 4)	4	axial vectors, ${\tau^{(4)}_1 \otimes \tau^{(1)}_4}$, e.g. ${\gamma_\mu\gamma_5}$
${\tau^{(6)}_1}$	(6, 1)	6	antisymmetric tensors, e.g. ${\sigma_{\mu\nu}}$

In the code, rep_label_list lists the 20 labels in this order, rep_dim_list their dimensions, and the dictionaries irrep_index, irrep_dim and irrep_texname map a label to its position, its dimension and its LaTeX name.

Tensor products and the Clebsch-Gordan series

The tensor product of two or more irreps of ${H(4)}$ is a representation of ${H(4)}$, but in general it is not irreducible (a few products are, e.g. those with a one-dimensional irrep, or ${\tau^{(2)}_1 \otimes \tau^{(4)}_1}$, which is an 8-dimensional irrep). It can be decomposed into a direct sum of irreps. This decomposition, called the Clebsch-Gordan series, has the form

$$D^{(\mu)} \otimes D^{(\nu)} = \bigoplus_\sigma a_\sigma \, D^{(\sigma)}, \tag{2}$$

where the symbol ${D^{(\mu)}}$ denotes a generic irrep ${\tau^{(k)}_l}$. The coefficient ${a_\sigma}$ is the number of times the irrep ${D^{(\sigma)}}$ appears in the decomposition of the tensor product, i.e. the multiplicity of ${D^{(\sigma)}}$.

For a given tensor product, the multiplicities can be computed from the characters of the irreps, ${\chi^{(\sigma)}(g) = \mathrm{Tr}\, D^{(\sigma)}(g)}$, which take the same value on all the elements of a conjugacy class:

$$a_\sigma = \left\langle \chi^{(\sigma)}, \chi^{(\mu)} \chi^{(\nu)} \right\rangle = \frac{1}{384} \sum_{i=1}^{20} k_i \, \chi^{(\sigma)}_i \, \chi^{(\mu)*}_i \, \chi^{(\nu)*}_i , \tag{3}$$

where ${384}$ is the number of elements of the group, the index ${i}$ runs over the 20 conjugacy classes of ${H(4)}$, ${k_i}$ is the number of elements in class ${i}$, and ${\chi^{(\sigma)}_i}$ is the character of the irrep ${\sigma}$ on class ${i}$. All the characters of ${H(4)}$ are real, so the complex conjugation can be dropped. For a product of more irreps, ${\chi^{(\mu)}_i \chi^{(\nu)}_i}$ is replaced by the product of all their characters. For the derivation of the formula and the related group theory, we refer the reader to chapter 4 of Jones (1990).

In the code. get_multiplicities gives the multiplicities ${a_\sigma}$ of a tensor product of any number of irreps, as a list in the order of rep_label_list, and latex_print_multiplicities gives the decomposition as a LaTeX string. The character table and the class orders (the number of elements in each class) are copied from Baake et al. (1982), and are available as char_table (one row per irrep, one column per class) and class_orders.

from h4lat import get_multiplicities, latex_print_multiplicities, rep_label_list

muls = get_multiplicities((4, 1), (4, 1))
print({rep_label_list[i]: m for i, m in enumerate(muls) if m})  # {(1, 1): 1, (3, 1): 1, (6, 1): 1, (6, 3): 1}
print(latex_print_multiplicities((4, 1), (4, 1)))

The next page, Clebsch-Gordan coefficients, shows how to find the basis in which this decomposition is explicit.

References

• M. Baake, B. Gemünden and R. Oedingen, Structure and representations of the symmetry group of the four-dimensional cube, J. Math. Phys. 23, 944–953 (1982); Erratum: J. Math. Phys. 23, 2595 (1982), doi:10.1063/1.525461.
• M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. E. L. Rakow, G. Schierholz and A. Schiller, Lattice operators for moments of the structure functions and their transformation under the hypercubic group, Phys. Rev. D 54, 5705–5714 (1996), doi:10.1103/PhysRevD.54.5705, arXiv:hep-lat/9602029.
• H. F. Jones, Groups, Representations and Physics (Adam Hilger, Bristol, 1990).

All the references of the wiki are collected on the References page.