Fundamental representations of sl3 ($V({\omega }_1)={\mathbb{C}}^3,V({\omega }_2)={\Lambda }^2{\mathbb{C}}^3$)

The two fundamental representations of ${\mathfrak{sl}}_3$ are the defining representation $V({\omega }_1)={\mathbb{C}}^3$ and its dual

The six smallest complex irreducible representations of SU(3) have highest weights $(0,0),(1,0),(0,1),(2,0),(0,2),(1,1)$ and dimensions $1,3,3,6,6,8$. Their Young diagram row lengths are $(0,0),(1,0),(1,1),(2,0),(2,2),(2,1)$. They are realized by scalars, defining vectors, covectors (equivalently antisymmetric tensors of degree two), upper and lower symmetric tensors of degree two, and mixed tensors $T^i{}_j$ with zero trace, respectively. The Weyl dimension formula is $\dim (p,q)=(p+1)(q+1)(p+q+2)/2$. For $p+q\ge 3$ this is at least ten, proving that the list exhausts the six smallest representations.

Split the two dual factors into antisymmetric and symmetric tensors. The invariant Levi-Civita symbol identifies the antisymmetric pair with $3$, giving $6\oplus \overline{3}$ after multiplying by the remaining triplet. On the symmetric part, contraction $T^a{}_{ba}$ maps onto $\overline{3}$; the equivariant splitting is $T^a{}_{bc}=({\delta }_b^a{v}_c+{\delta }_c^a{v}_b)/4$. Its trace-free kernel has dimension $18-3=15$ and highest weight with Dynkin labels $(1,2)$. The Weyl dimension formula gives dimension $15$, so this is exactly the irreducible kernel. It is conjugate to the other fifteen-dimensional module with labels $(2,1)$; dimension alone does not distinguish the two.

The tensor product of Lie algebra representations $3\otimes 3$ splits into the symmetric $6$ and antisymmetric $\overline{3}$. Then $6\otimes 3=10\oplus 8$ and $\overline{3}\otimes 3=8\oplus 1$, as highest-weight character subtraction verifies. The decuplet is fully symmetric under quark permutations, the singlet fully antisymmetric, and the two octets constitute the two-dimensional mixed permutation multiplicity space. Applying the Pauli constraint on three-quark flavour multiplets gives one ground-state spin-one-half octet and one spin-three-halves decuplet.

// Target: physics.bigb

For the special linear Lie algebra ${\mathfrak{sl}}_3(\mathbb{C})$ with upper triangular positive roots, the simple coroots are $\mathrm{diag}(1,-1,0)$ and $\mathrm{diag}(0,1,-1)$. Their dual fundamental weights are ${\omega }_1(\mathrm{diag}(t_1,t_2,t_3))=t_1$ and ${\omega }_2(\mathrm{diag}(t_1,t_2,t_3))=t_1+t_2=-t_3$. In simple-root coordinates, ${\omega }_1=(2{\alpha }_1+{\alpha }_2)/3$ and ${\omega }_2=({\alpha }_1+2{\alpha }_2)/3$. The primitive vectors $e_1\in {\mathbb{C}}^3$ and $e_1\wedge e_2\in {\Lambda }^2{\mathbb{C}}^3$ realize these Fundamental representations of sl3.

For the defining representation $V=V({\omega }_1)$ of ${\mathfrak{sl}}_3$,