Orthogonal Colour Tensors Overview

• Orthogonal colour tensors are multidimensional arrays with mode-wise orthogonal symmetries and distinct colour assignments that facilitate independent transformation actions.
• They provide a unified framework for invariant and representation theory, enabling systematic tensor contractions and symmetry classification in fields like gauge theory and graph theory.
• They support algorithmic computations through coloured Brauer diagrams and tensor decompositions, impacting areas such as numerical multilinear algebra and image processing.

Orthogonal colour tensors are multidimensional arrays endowed with both mode-wise orthogonal symmetries and colour-label assignments that index independent symmetry actions. Arising in invariant theory, representation theory, quantum field theory, numerical multilinear algebra, and graph theory, orthogonal colour tensors provide a unified framework for the description, classification, and computation with tensorial objects exhibiting distinct orthogonal invariances or colour/representation structures.

1. Formal Definition and Symmetry Structure

An orthogonal colour tensor is a tensor $$\mathcal T \in \mathbb R^{p_1 \times \cdots \times p_d}$$ (or over $\mathbb C$), where each index $i=1,\ldots,d$ is assigned a “colour,” and the tensor admits a symmetry group

$H = O(p_1) \times \cdots \times O(p_d)$

acting independently on each mode: for $A_i \in O(p_i)$,

$$(A_1, \dots, A_d) \cdot \mathcal T = \mathcal T \times_1 A_1 \times_2 \cdots \times_d A_d,$$

where $\times_m$ denotes $m$-mode multiplication. Each “colour” refers to a particular mode’s orthogonal action. In quantum field theory (QFT) and combinatorics, “colour” generalizes to representation-theoretic labels, e.g., fundamental or adjoint indices in gauge theory, each acted on by an independent orthogonal or unitary group factor (Tokcan et al., 2020, Bourjaily et al., 29 Dec 2025).

Polynomials or multilinear functions built from such tensors are called orthogonal colour invariants if they are fixed under the full group $H$ action (Williams, 2012, Tokcan et al., 2020). The symmetries enforced by $H$ ensure that both algebraic (index contraction invariance) and geometric (rotational/reflectional) properties are preserved within each colour.

2. Invariant Theory: Coloured Brauer Diagrams and Contraction Patterns

The full structure of $H$-invariant polynomials is codified combinatorially. Invariant theory establishes that the algebra of $H$-invariants is generated by specific complete contraction monomials, each corresponding to a coloured Brauer diagram (Tokcan et al., 2020, Williams, 2012).

Let $m$ be even with $m = 2e$. Then, for $d$ colours, each invariant of degree $m$ is determined by a $d$-tuple of perfect matchings on $\{1,\dots,m\}$ — one per colour — yielding a $d$-edge-coloured, $d$-regular graph (coloured Brauer diagram). Explicitly, for a diagram $\mathbf D$:

$$\mathcal P_{\mathbf D}(\mathcal T) = \sum_{(\text{all indices})} \prod_{u=1}^m t_{i^u_1, \ldots, i^u_d} \prod_{(u, v) \text{ edge of colour } c} \delta_{i^u_c, i^v_c}$$

(Tokcan et al., 2020). The degree-$m$ invariants form a basis labeled by coloured Brauer diagrams of size $m$.

For generic $n_i \geq m$, the dimension of the degree-$m$ invariants is $((m-1)!!)^d$ (Tokcan et al., 2020). For the product of $r$ orthogonal groups $O(n_1)\times\cdots\times O(n_r)$ acting on $$V=\mathbb R^{n_1}\otimes\cdots\otimes \mathbb R^{n_r}$$, dimension and explicit bases are controlled combinatorially via the enumeration and classification of $r$-tuples of matchings up to simultaneous conjugation by $S_{2m}$ (Williams, 2012).

This construction generalizes the First Fundamental Theorem (FFT) of invariants for single orthogonal groups to coloured circumstances, and the graphical calculus yields algorithmic partial contraction schemes for evaluating such invariants.

3. Orthogonality in Gauge Theory Colour Tensors

In perturbative gauge theories, colour dependence of scattering amplitudes is encoded in tensors built from (possibly arbitrarily nested) Clebsch-Gordan coefficients, structure constants $f^{abc}$, and traces. The construction of a complete, linearly independent, and mutually orthogonal basis for all colour structures is achieved by associating a planar trivalent tree to the process, with each internal edge labeled by an irreducible representation, and each vertex by an orthonormal basis of Clebsch-Gordan tensors (Bourjaily et al., 29 Dec 2025).

Let $\mathfrak{g}$ denote a (semi-)simple Lie algebra, and $\{\mathbf{R}_i\}_{i=1}^n$, $\{\mathbf{T}_j\}_{j=1}^m$ the incoming and outgoing representations (colours) for an $n\to m$ amplitude. The basis tensors $T_\alpha$ are constructed by:

• Fixing a trivalent tree graph $\Gamma$ connecting all external legs;
• Labeling each internal edge by an irrep and each vertex by a multiplicity index of the corresponding intertwiner (Clebsch-Gordan tensor);
• Contracting all internal indices to define $T_\alpha$ on the external indices.

These tensors are equipped with an inner product defined by index contraction against the invariant metrics on each representation. The orthonormality of Clebsch-Gordan coefficients and application of Schur’s lemma ensure that the Clebsch-trivalent tree basis is automatically orthogonal:

$$\langle T_\alpha, T_\beta \rangle = \delta_{\alpha \beta}$$

(Bourjaily et al., 29 Dec 2025).

General colour factors (e.g., from Feynman diagrams) can be reduced to this orthonormal basis via local “bubble deletion," "F-moves," and "R-moves”—i.e., diagrammatic manipulations following group-theoretic and tensor contraction rules.

4. CSA, Decomposition Algorithms, and Image Processing Applications

Orthogonal colour tensors are also central in multilinear numerical frameworks, such as the tensor CS-decomposition (T-CSD) and orthogonal CP decompositions. The T-product framework defines the notion of T-orthogonality for 3-tensors $\mathcal Q \in \mathbb R^{n\times n \times n_3}$:

$$\mathcal Q^T*\mathcal Q = \mathcal Q*\mathcal Q^T = \mathcal I$$

where $*$ is the T-product—diagonalization in the transform domain—and $\mathcal Q^T$ transposes and reverses frontal slices (Zhang et al., 2021).

The T-CS decomposition factorizes such an orthogonal tensor into spatial and colour-basis orthogonal factors:

$$[\mathcal Q_1;\mathcal Q_2] = [\mathcal U \oplus \mathcal V] * [\mathcal C; \mathcal S] * \mathcal Z^T$$

with orthogonality preserved in both the spatial and colour dimensions; $\mathcal Z \in \mathbb R^{3\times 3\times 3}$ (for colour images) is an orthogonal colour-basis transform (Zhang et al., 2021). These methods allow direct manipulation of multi-channel (e.g., RGB) images under orthonormal spatial and colour transformations, and share theoretical consistency with classical matrix decompositions.

Optimized computational methods—such as the augmented Lagrangian method for orthogonal CP decomposition—yield best low orthogonal rank approximants to a given tensor, guaranteed by the lower semicontinuity of orthogonal rank (Zeng, 2021).

5. Second-Order Geometric Invariants and Divergence-Free Colour Tensors

In differential geometry, second-order natural tensors (functions of a metric and its first and second derivatives that commute with diffeomorphisms) are classified via their correspondence to $O(n)$-invariant contraction patterns (“colour tensors”) among curvature tensors and the metric (Navarro, 2013). The main result establishes:

• Every divergence-free, second-order natural $p$-tensor corresponds to an $O(g_x)$-invariant source, explicitly as a linear combination of complete contraction patterns between the free and curvature indices via the inverse metric;
• For rank-2 symmetry, the unique sequence of Lovelock tensors spans all such invariants. These are generated by contracting up to $k$ Riemann tensors (viewed as normal 2-tensors) and $(n-2k-1)$ metrics.

Thus, the entire ring of divergence-free, second-order invariant tensors is governed by orthogonal colour contraction rules (Navarro, 2013).

6. Orthogonal Colourings in Graph Theory

Orthogonal colourings also arise in graph theory, particularly in the study of tensor (Kronecker) products of graphs and their chromatic properties. A $k$-orthogonal colouring of a graph is a sequence of $k$ proper colourings that are pairwise orthogonal: if two vertices share the same colour in one colouring, they must differ in all others. The problem of determining perfect $k$-orthogonal colourings reduces to subgraph embeddings into products of complete graphs, and the minimum required number of colours in the tensor-product graph admits tight upper bounds based on the factors' chromatic numbers (MacKeigan, 2020).

Orthogonality underlies the construction of multi-factor graph invariants and reflects the algebraic underpinnings found in tensor invariant theory.

7. Enumeration, Complexity, and Combinatorial Structures

The enumeration of orthogonal colour tensor invariants is governed by matchings and coloured graph factorization. For the degree-$2m$ component of the invariant algebra under $r$ orthogonal actions, the dimension is

$$\dim P_{2m}(V)^K = \frac{1}{(2m)!} \sum_{\sigma\in S_{2m}} N(\text{ct}(\sigma))^r = \sum_{\lambda \vdash 2m} \frac{N(\lambda)^r}{z_\lambda}$$

where $N(\lambda)$ counts matchings commuting with a given cycle-type, and $z_\lambda$ is the centralizer size. This combinatorics connects to the enumeration of r-regular edge-coloured graphs, Brauer algebras, and phylogenetic trees (Williams, 2012, Tokcan et al., 2020).

Algorithmic evaluation leverages partial contractions in each colour, typically scaling polynomially in tensor size and degree for small degrees, and practical schemes for computation are provided (Tokcan et al., 2020).

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Orthogonal colour tensors form a central structural concept cutting across geometry, combinatorics, algebra, numerical analysis, and physics, built on rigorous orthogonal invariance under independent colour-labelled symmetries. Their complete characterization via contraction graphs and orthogonal bases underpins their utility for both theoretical classification and practical computation (Tokcan et al., 2020, Williams, 2012, Bourjaily et al., 29 Dec 2025, Zeng, 2021, Zhang et al., 2021, Navarro, 2013, MacKeigan, 2020).