Implementation notes

natto computes the operators of Wen, 2026, but not always by the paper’s formulas taken literally. This page collects the rewritings it uses instead. Each leaves the computed quantity unchanged and only makes it cheaper. Equations of the paper are written (W15) for Eq. 15 of Wen, 2026.

Notation¶

$n$ is the rank of the Cartesian tensor and $\ell$ the weight of an ICT. The reduction is built from the rank-lowering tensors $F^{(p)}$ (W2)–(W5) and the natural projector $\mathsf E^{(\ell)}$ (W7), which give the mappings, their Gram matrix and their duals,

G^{(p)} = \mathsf E^{(\ell)} F^{(p)}, \qquad g_{pq} = \frac{1}{2\ell+1} \langle G^{(p)}, G^{(q)} \rangle, \qquad \widetilde G^{(p)} = \sum_q (g^{-1})_{pq}\, G^{(q)} ,

(1)

(W13), (W15) and (W16), where $\langle \cdot, \cdot \rangle$ contracts every index. The composed operator $S^{(p)} = G^{(p)} \cdot \widetilde G^{(p)}$ contracts the ICT indices of a mapping and its dual.

The natural projector as a sum over matchings¶

Each term of (W7) is one perfect matching $m$ of the projector’s $2\ell$ indices, and its coefficient depends only on the number $t(m)$ of pairs within each index group:

\mathsf E^{(\ell)} = \sum_{m} \frac{c_{t(m)}}{N_{t(m)}}\, D_m, \qquad c_t = -\frac{(\ell-2t+2)(\ell-2t+1)}{2t\,(2\ell-2t+1)}\, c_{t-1}, \quad c_0 = 1,

(2)

with $D_m$ the product of the matching’s deltas and $N_t$ the number of matchings with that $t$ . The recursion is (W S27). The projector has $(2\ell-1)!!$ terms, so a product of two projectors has $[(2\ell-1)!!]^2$ term pairs, about 10⁸ at $\ell = 6$ . The rewritings below avoid forming such products.

One projector suffices¶

The projector is symmetric and idempotent, so one of the two projectors in an inner product of mappings can be left out:

\langle G^{(p)}, G^{(q)} \rangle = \langle F^{(p)}, G^{(q)} \rangle, \qquad g_{pq} = \frac{1}{2\ell+1} \langle F^{(p)}, G^{(q)} \rangle .

(3)

The same holds for the composed operator, whose tensor indices stay free:

S^{(p)}_{\boldsymbol A \boldsymbol B} = \sum_q (g^{-1})_{pq}\, F^{(p)}_{\boldsymbol\sigma \boldsymbol A}\, G^{(q)}_{\boldsymbol\sigma \boldsymbol B} .

(4)

Both cost $(2\ell-1)!!$ products instead of $[(2\ell-1)!!]^2$ .

Mappings as vectors¶

Every mapping of a weight, whether a candidate, a dual or a symmetry-adapted mapping, is the projector applied to a rational combination of the same rank-lowering tensors $F_0, \dots, F_{N-1}$ , so it is stored as a coefficient vector $\boldsymbol c \in \mathbb Q^N$ . All inner products then come from one table, computed once per weight and rank:

L_{ij} = \langle F_i, \mathsf E\, F_j \rangle, \qquad \bigl[\langle \cdot, \cdot \rangle\bigr] = C_1\, L\, C_2^{\mathsf T},

(5)

where the rows of $C_1$ and $C_2$ are the coefficient vectors of two lists of mappings. The Gram matrix of any mappings, the adapted Gram matrix $g^Q = M g M^{\mathsf T}$ , and the orthonormal mappings $\widehat G = g^{-1/2} G$ (W21) are all matrix algebra on these vectors.

Permuted mappings¶

Permuting the tensor indices of a rank-lowering tensor gives another rank-lowering tensor, up to the sign of reordering its Levi-Civita symbol:

P\, F_i = s_i\, F_{\pi(i)}, \qquad s_i = \pm 1 .

(6)

So a permuted mapping is another coefficient vector, and the mixing matrix of a symmetry generator (W30),

M = g^{-1} O, \qquad O_{pq} = \frac{1}{2\ell+1} \langle G^{(p)}, P\, G^{(q)} \rangle ,

(7)

is read from the table (5) with no contraction.

Contracting products of deltas and Levi-Civita symbols¶

Every operator is a sum of products of Kronecker deltas and Levi-Civita symbols, so contracting operators reduces to contracting such products term by term, with no arrays. Following the repeated indices, a chain of deltas collapses to one delta, a closed chain is a factor of 3, and a Levi-Civita symbol with a repeated index vanishes:

\delta_{a i_1}\, \delta_{i_1 i_2} \cdots \delta_{i_k b} = \delta_{ab}, \qquad \delta_{i_1 i_2} \cdots \delta_{i_k i_1} = 3, \qquad \varepsilon_{i i c} = 0 .

(8)

Two Levi-Civita symbols joined by a repeated index are expanded into deltas by their determinant identity, and the chains are followed again:

\varepsilon_{abc}\, \varepsilon_{def} = \sum_{\pi \in S_3} \operatorname{sgn}(\pi)\, \delta_{a\pi(d)}\, \delta_{b\pi(e)}\, \delta_{c\pi(f)} .

(9)

Two symbols with no index in common are kept as they are, since expanding them would turn one product into six.

Full contraction is cycle counting¶

Every entry of the table is a full contraction, a number rather than an operator. A product of deltas in which every index occurs twice is a union of cycles, each a factor of 3, and with one Levi-Civita symbol on each side the paths between the two symbols give the sign:

\prod_k \delta_{u_k v_k} = 3^{\operatorname{cyc}}, \qquad \varepsilon_{i_1 i_2 i_3}\, \varepsilon_{j_1 j_2 j_3} \prod_k \delta_{u_k v_k} = 6\, \operatorname{sgn}(\pi)\, 3^{\operatorname{cyc}} ,

(10)

where $\pi$ is the bijection the delta paths make between the slots of the two symbols, and the product is zero if a path joins two slots of the same symbol. The table is filled by counting cycles, without forming any terms.

Selecting independent mappings¶

The number of independent mappings of weight $\ell$ in a rank- $n$ tensor, Table II of Wen, 2026, follows from $j \otimes 1 = (j-1) \oplus j \oplus (j+1)$ :

N(n, \ell) = N(n-1, \ell+1) + [\ell \ge 1]\, \bigl(N(n-1, \ell-1) + N(n-1, \ell)\bigr), \qquad N(0, \ell) = \delta_{\ell 0} .

(11)

The candidates are scanned in order, and the scan stops once $N(n, \ell)$ are kept. A candidate with Gram entries $b$ against the kept mappings and $d$ with itself is kept exactly when

d - b^{\mathsf T} g^{-1} b \neq 0 ,

(12)

with $g^{-1}$ extended by the bordered inverse as candidates are kept. The decision is exact, where the paper’s Algorithm 1 decides by a pivoted QR against a tolerance.

References¶

Wen, M. (2026). Reusable Operators for Irreducible Cartesian Tensor Decomposition and Coupling. arXiv Preprint arXiv:2609.05971. 10.48550/arXiv.2609.05971