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Color Octet

The eight possible color combinations of quark charge, forming the basis of the QCD color symmetry group.
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The statement of the theorem

Let TaT^a be the generators of SU(3)cSU(3)_c. The color octet state ∣octet⟩|\text{octet}\rangle is a state vector in the adjoint representation, transforming under the 88-dimensional irreducible representation of SU(3)cSU(3)_c. The basis states are orthogonal to the singlet and triplet states. The Casimir invariant C2C_2 for the octet representation is C2(octet)=C2(fund)+C2(adj)−C2(fund)=3C_2(\text{octet}) = C_2(\text{fund}) + C_2(\text{adj}) - C_2(\text{fund}) = 3. Specifically, the generators satisfy Tr(TaTb)=12δab\text{Tr}(T^a T^b) = \frac{1}{2} \delta^{ab} for the fundamental representation, and the octet generators ToctetaT^a_{\text{octet}} satisfy Tr(ToctetaToctetb)=12δab\text{Tr}(T^a_{\text{octet}} T^b_{\text{octet}}) = \frac{1}{2} \delta^{ab}.