Symmetric Cube of SU(3)

• The symmetric cube of SU(3) is the totally symmetric third power of the fundamental representation, decomposing into a decuplet, octets, and a singlet.
• It underpins the computation of Clebsch–Gordan coefficients and scalar factors, with methodologies leveraging occupation-number representations and SU(2) subgroup couplings.
• Its applications span baryon multiplet classification, unified gauge theories, and SU(3)-symmetric quantum chains, linking abstract theory to observable phenomena.

The symmetric cube of SU(3) is the totally symmetric third power of the fundamental representation, corresponding to combining three triplets in a symmetric fashion. This object underpins several theoretical frameworks: the classification of baryonic multiplets in particle physics, the construction and computation of Clebsch–Gordan (CG) coefficients and scalar factors, connections to flavor symmetries and unified field theory models, and symmetry-protected phenomena in quantum chains. Across group-theoretical, algebraic, and quantum many-body contexts, the symmetric cube links the microstructure of SU(3) to rich physical applications.

1. Definition and Representation-Theoretic Structure

The symmetric cube of SU(3) refers to the symmetrized tensor product $\text{Sym}^3(\mathbf{3})$ of the fundamental representation. When combining three triplets, the SU(3) representation decomposition is: $$\mathbf{3} \otimes \mathbf{3} \otimes \mathbf{3} = \mathbf{10} \oplus \mathbf{8} \oplus \mathbf{8} \oplus \mathbf{1}$$ where $\mathbf{10}$ is the totally symmetric decuplet (Young tableau: three boxes in a row), $\mathbf{8}$'s are octets with mixed symmetry, and $\mathbf{1}$ is the singlet. The symmetric cube is, by construction, associated with the $\mathbf{10}$ representation.

The construction of basis states and corresponding CG coefficients for the symmetric cube relies on permutational symmetries and weight decompositions in SU(3). As described in (Martins et al., 2019), the use of occupation-number representations and harmonic oscillator basis states allows each weight vector to be naturally realized as a symmetrized triple product, with Weyl group permutations giving efficient symmetry relations among the coefficients.

2. Clebsch–Gordan Coefficients and Scalar Factor Decompositions

The CG coefficients for SU(3) encode how composite states arising from the tensor product of three triplets transform under SU(3) symmetry. For the symmetric cube, these coefficients specify the projection onto the $\mathbf{10}$ states and more generally onto all irreducible components.

Explicit formulas provided in (Garcia-Recio et al., 2010) and (Martins et al., 2019) use the reduction of SU(n) CG coefficients into products of subgroup coefficients (e.g., SU(6)$\supset$SU(3)$\times$SU(2)), organized by scalar factors: $$|R; \mu\rangle = \sum_{m_1,m_2,m_3} \langle 3, m_1; 3, m_2; 3, m_3\,|\;R, \mu\rangle \;\;|3, m_1\rangle|3, m_2\rangle|3, m_3\rangle$$ In practice, tables for scalar factors (phase-conventioned reduced matrix elements) such as Table II in (Garcia-Recio et al., 2010) and phase relations between subgroups (Eqs. (D1)–(D8)) facilitate full reconstruction of CG coefficients for the symmetric cube across embedding and breaking chains.

A distinctive feature in (Martins et al., 2019) is that starting from an SU(2)-adapted occupation-number basis enables direct computation of SU(3) coefficients via SU(2) CGs and higher $6j$, $9j$ symbols, with Weyl reflections relating the coefficients for permuted weights. Multiplicity-free decomposition—each irreducible occuring only once—applies in the symmetric cube case, reducing algebraic complexity.

3. Symmetric Cube in Spin–Flavor Symmetric Schemes

The role of the symmetric cube is pronounced in spin–flavor symmetry, especially in baryon composition. In the SU(6) spin–flavor scheme, baryons are encoded in the 56-plet, which decomposes to flavor SU(3) and spin SU(2) multiplets. The symmetric flavor sector (three quarks symmetrized in flavor), realized as the $\mathbf{10}$ decuplet from $\text{Sym}^3(\mathbf{3})$, is central to the identification of baryonic states.

Scattering amplitude calculations and allowed decay channels directly depend on the symmetry structure of the CG coefficients for the symmetric cube ((Garcia-Recio et al., 2010), section 3). Certain channels are enhanced or forbidden by the symmetric and antisymmetric content of SU(3) products, with scalar factors determining the strength of allowed couplings.

4. Algebraic Realizations: Coupled SU(2) Subgroups and Octonion Splitting

The construction of SU(3) as a coupled system of three SU(2) subgroups enables an algebraic model of the symmetric cube (Raphaelian, 2016, Pushpa et al., 2011). In this formalism, each quantization axis (associated with I-spin, U-spin, and V-spin) defines a local SU(2) symmetry. The algebraic coupling framework yields intrinsic angular momentum eigenvectors—symmetrized or antisymmetrized across axes—corresponding to the various SU(3) representations.

Internal geometric phases dictate the cyclic ordering and exchange symmetry among axes, with basis state configurations reflecting structures observed in color-charge and electroweak phenomena ((Raphaelian, 2016), section 3). The octonion splitting, detailed in (Pushpa et al., 2011), maps the SU(3) generators onto octonion units, connecting the underlying non-associative algebra to flavor multiplets. For instance, the construction allows the identification of symmetric cube states through nested applications of SU(2) ladder operators associated with the three sub-algebras.

5. Symmetric Cube in Grand Unified and Extended Gauge Group Models

In unified model-building, the symmetric cube appears in the context of embedding the Standard Model fermions into representations of higher SU(3) powers. The [SU(3)]³ group, as analyzed in (Rodríguez et al., 2016), encapsulates the full particle content in a "symmetric cube" of SU(3)—formally, the 27-dimensional representation obtained from combining three SU(3) factors.

The symmetric structure determines electric charge assignments (with explicit operators: $$Q = I_{L3} + (1/\sqrt{3})I_{L8} + c_X X_{R3} + (2d_X/\sqrt{3}) X_{R8}$$), photon null-space in the gauge boson mass matrix, and patterns of neutral gauge bosons $Z', Z''$. Stringent constraints on $Z'$ masses ($M_{Z'}>2.5$ TeV) and mixing angles ($10^{-3}$ radians) reflect the precision with which the symmetric cube symmetry constrains beyond-SM physics.

The mathematical machinery—orthogonal rotations between basis states, phase conventions, and scalar factor tables—are directly applied to the analysis of experimental signals, linkage to E₆ representation content, and model discrimination ((Rodríguez et al., 2016), sections 3–5).

6. Quantum Many-Body Systems and Universality Classes

In quantum chains with SU(3) symmetry, the symmetric cube governs critical phenomena and universality classes. The pure trimer (PT) point in the Dimer-Trimer spin-1 chain (Mashiko et al., 2020) is an SU(3) symmetric point corresponding to an exactly solvable model with central charge $c=2$ and scaling dimension $x=2/3$, matching the level-1 SU(3) WZW conformal field theory. The transition at PT is Berezinskii-Kosterlitz-Thouless (BKT)-like, with correlation length scaling: $\xi \sim \exp[C(\theta_C-\theta)^{-\sigma}]$ and excitation energy corrections: $$\Delta E_{st}(\pm 2\pi/3) = \frac{2\pi v_0}{N}\left[x_{st}+\frac{d_{st}}{\ln(N/N_0)}\right]$$ Numerical diagonalization confirms the role of totally symmetric states and their criticality, with entanglement entropy and spectrum diagnostics reinforcing the correspondence to symmetric cube representations.

Edge states in the two-box symmetric SU(3) chain (Nataf et al., 2021) further link the representation theory to observable phenomena: localized adjoint edge states inherit anomalous scaling $\Delta \sim 1/(N_s \log N_s)$, dominating entanglement measures. Screening these edge states by coupling adjoint spins restores universal bulk values ($c \approx 2$), corroborating the SU(3)₁ WZW classification.

7. Related Decompositions and Symmetry Breaking Chains

The computation of scalar factors for decompositions

$$\mathrm{SU}(4) \supset \mathrm{SU}(3)\times\mathrm{U}(1), \quad \mathrm{SU}(3) \supset \mathrm{SU}(2)\times\mathrm{U}(1)$$

enables the mapping of symmetric cube states onto isospin and hypercharge sectors. This is necessary for translating theoretical symmetry assignments into experimentally relevant particle quantum numbers ((Garcia-Recio et al., 2010), section 4).

The symmetry under permutations in the symmetric cube ensures that flavor wave functions of composite particles (e.g., baryons) remain invariant, and scalar factors provide the dictionary for tracing these symmetries through the breaking chain down to physical multiplets.

Summary

The symmetric cube of SU(3) is a central organizing principle for the construction of SU(3) irreducible representations, the computation of CG coefficients and scalar factors, the algebraic modeling of coupled SU(2) subgroups, the embedding and decomposition in unified gauge models, and the analysis of critical phenomena in SU(3)-symmetric quantum chains. Across all contexts, the symmetrization of three triplets yields states whose group-theoretic, algebraic, and physical properties manifest in both abstract representation theory and concrete applications to meson and baryon interactions, unified model constraints, and definitive signatures in quantum many-body systems. The structure and computational tools—scalar factor tables, occupation-number basis, SU(2)-inherited formalism, phase conventions, and Weyl group symmetries—provide a unified framework for analyzing the symmetric cube and exploiting its rich implications in contemporary theoretical physics.