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A simple-root system is a fundamental system of a root system: a basis $\Delta$ consisting of roots such that every root has coordinates all nonnegative or all nonpositive. Its nonnegative linear combinations select the associated positive roots. For a reduced crystallographic root system, these simple roots are precisely the positive roots that cannot be written as the sum of two positive roots; every root is then an integer combination of simple roots with coefficients of one sign.

For the A2 root system, take ${\alpha }_1=e_1-e_2$ and ${\alpha }_2=e_2-e_3$. The third positive root is ${\alpha }_1+{\alpha }_2$, so it is not simple. In a complex semisimple Lie algebra, the simple roots select raising and lowering generators through their root vectors, while their coroots define the Dynkin labels of a highest weight.

For the defining three-dimensional special linear Lie algebra representation $V$, contraction $(uv)\otimes f\mapsto f(u)v+f(v)u$ maps $(S^{2}V)\otimes V^{\ast }$ onto $V$. The kernel contains the highest-weight vector $e_1^2\otimes e_3^{\ast }$ of Dynkin labels $(2,1)$. The Weyl dimension formula gives its irreducible module dimension $15$, the entire kernel dimension; the Weyl complete reducibility theorem supplies the displayed splitting.

For the complex special linear Lie algebra ${\mathfrak{sl}}_3$, tensoring an irreducible highest-weight representation by the defining representation gives the displayed sum, omitting terms with negative Dynkin labels. It follows by multiplying its Weyl character formula by $x_1+x_2+x_3$. In particular ${\Gamma }_{2,1}\otimes {\Gamma }_{1,0}={\Gamma }_{3,1}\oplus {\Gamma }_{1,2}\oplus {\Gamma }_{2,0}$, with dimensions $24+15+6=45$.

For the special linear Lie algebra ${\mathfrak{sl}}_3$, the irreducible highest-weight representation with Dynkin labels $(a,b)$ has the displayed formal character on diagonal matrices with $x_1{x}_2{x}_3=1$. The interlacing rows $(a+b,b,0)$, $(p,q)$ and $(r)$ enumerate a Gelfand–Tsetlin basis; each triple contributes the exponent differences $(r-(p+q-r),(p+q-r)-(a+2b-p-q))$ as its weight. This gives exact integer weight multiplicities, including repeated weights, without floating-point character evaluation.

With $H_1=\mathrm{diag}(1,-1,0)/2$ and $H_2=\mathrm{diag}(1,1,-2)/(2\sqrt{3})$, the simple roots are $(1,0)$ and $(-1/2,\sqrt{3}/2)$. Their coroots give Dynkin labels $a=2p$ and $b=\sqrt{3}q-p$ for Cartesian highest-weight coordinates $(p,q)$. Finite-dimensional irreducible representations require $a,b\in {\mathbb{Z}}_{\ge 0}$. The vertical coordinate is $\sqrt{3}/2$ times flavor hypercharge.

For generators normalized by ${\mathrm{tr}}_3(t_a{t}_b)={\delta }_{ab}/2$, the quadratic Casimir operator on the irreducible SU(3) representation of Dynkin labels $(a,b)$ has the displayed scalar value. In SU(3) highest-weight coordinates $(p,q)$ it is $p^2+q^2+p+\sqrt{3}q$.

To derive this, write the operator as $H_1^2+H_2^2+{\sum }_{i=1}^3(E_{+}^i{E}_{-}^i+E_{-}^i{E}_{+}^i)$. On a highest-weight vector, the raising operators vanish, and the three root commutators sum to $H_1+\sqrt{3}{H}_2$. Centrality and the Schur lemma extend this value to the representation. The singlet, fundamental, adjoint and decuplet values are $0,4/3,3,6$, respectively. Tensor labels and multiplet dimensions are summarized in the Particle Data Group's SU(n) multiplet guide.

The summands have Dynkin labels $(2,1)$, $(0,2)$ and $(1,0)$. Convolving the triplet and octet weight diagrams and applying highest-weight character subtraction removes these three characters in that order. Their dimensions $15+6+3=24$ match the product.

// Target: quantum-field-theory.bigb

The degree-three monomials $z_1^a{z}_2^b{z}_3^c$, with $a+b+c=3$, form a basis of this ten-dimensional highest-weight representation. Its highest monomial is $z_1^3$, and $E_{21}=z_2{\partial }_{z_1}$ and $E_{32}=z_3{\partial }_{z_2}$ generate every basis monomial. Its Dynkin labels are $(3,0)$, and the weight of a monomial is $(a-b,b-c)$. In coordinates $(h_1,(h_1+2h_2)/\sqrt{3})$, the ten weights form four horizontal rows of lengths $4,3,2,1$, all with weight multiplicity one. The baryon decuplet realizes this representation of approximate flavor symmetry.

// Target: special-relativity.bigb

For the special linear Lie algebra $\mathfrak{sl}(3)$, with Dynkin labels indexing irreducible highest-weight representations,The summands have dimensions $60,24,15,15,6$, summing to $120=15\cdot 16/2$. The identity follows from the formal character of a weight module formula ${\chi }_{S^{2}W}(z)=({\chi }_W(z{)}^2+{\chi }_W(z^2))/2$ and the Weyl character formula.

The central element $\omega I$ of SU(3), with $\omega =e^{2\pi i/3}$, acts on an irreducible representation of Dynkin labels $(a,b)$ as ${\omega }^{a+2b}$. This residue is its triality. In a tensor realization the representation has $a+2b$ fundamental boxes, and a complete column of three is a singlet; removing such columns preserves the residue. Tensor products add trialities and complex conjugation reverses them.

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.

A coordinate matrix expressing restriction of weights to the Cartan subalgebra of a Lie subalgebra. With Dynkin labels as column vectors, its rows encode the subalgebra's coroots in terms of the original coroots. It need not be an idempotent projection operator: an equal-rank embedding can give an invertible matrix. Applying it to the full weight system, including weight multiplicities, permits reconstruction of the restricted irreducible representations.