Representations of ${\mathfrak{sl}}_3\mathbb{C}$

Intuition [Beginner]

The Lie algebra ${\mathfrak{sl}}_3\mathbb{C}$ is the next step after ${\mathfrak{sl}}_2\mathbb{C}$, and its representation theory is the first place where the ladder picture of rank one gives way to a genuinely two-dimensional weight picture. Instead of a single linear chain of weight spaces, the weights of an irreducible representation of ${\mathfrak{sl}}_3\mathbb{C}$ sit at the points of a hexagonal grid in a plane. The convex hull of these points is a hexagon, and the symmetries of the hexagon are exactly the Weyl group of the root system $A_2$.

There are two Cartan generators, $H_1$ and $H_2$, that play the role of $H$ in the rank-one story. They commute, so every irreducible representation can be simultaneously diagonalised: each vector has a definite pair of eigenvalues, called its weight, drawn in a plane. Six raising and lowering operators move between weight spaces in six directions — the six roots of $A_2$, arranged like the spokes of a regular hexagon.

Two representations stand out as building blocks. The standard representation $V={\mathbb{C}}^3$ has weights at the three vertices of an upward-pointing triangle. The dual representation $V^{\ast }={\mathbb{C}}^3$ has weights at the three vertices of a downward-pointing triangle. Every other irreducible is built from tensor products of $V$ and $V^{\ast }$, indexed by two non-negative integers $(a,b)$ counting the symmetric powers in each direction. The dimension grows as a polynomial of degree three in $(a,b)$, and the adjoint representation $V_{1,1}$ of dimension eight (the famous "eight-fold way" of particle physics) is the rank-two analogue of the spin-one rung of ${\mathfrak{sl}}_2$.

Visual [Beginner]

A picture of a hexagonal weight diagram for the adjoint representation of ${\mathfrak{sl}}_3\mathbb{C}$. Six outer dots sit at the vertices of a regular hexagon, each labelled by a root vector: $\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2)$. A double dot at the centre indicates the multiplicity-two zero weight. Arrows along the six edge directions show the action of the raising and lowering operators $E_{{\alpha }_i}$ and $F_{{\alpha }_i}$. To the side, two smaller triangles label the two fundamental representations $V$ (upward triangle, weights $L_1,L_2,L_3$) and $V^{\ast }$ (downward triangle, weights $-L_1,-L_2,-L_3$).

The hexagon is the rank-two replacement for the linear ladder. Each irreducible representation $V_{a,b}$ has weights filling a hexagon (or a triangle, in the boundary cases $a=0$ or $b=0$) with multiplicities determined by how far inside the hexagon a weight lies — the inner layers stack up multiplicities while the outermost layer always has multiplicity one.

Worked example [Beginner]

Build the adjoint representation $V_{1,1}$ of ${\mathfrak{sl}}_3\mathbb{C}$ explicitly. The space is ${\mathfrak{sl}}_3\mathbb{C}$ itself, the eight-dimensional space of traceless $3\times 3$ matrices, and the action is the commutator: $X\cdot Y=[X,Y]$ for $X\in {\mathfrak{sl}}_3\mathbb{C}$ and $Y\in {\mathfrak{sl}}_3\mathbb{C}$.

Step 1. Pick a basis of the Cartan subalgebra. Set $H_1=\mathrm{diag}(1,-1,0)$ and $H_2=\mathrm{diag}(0,1,-1)$. These commute: $[H_1,H_2]=0$.

Step 2. Pick basis vectors for the six off-diagonal directions. Write $E_{ij}$ for the matrix with a 1 in the $(i,j)$ slot and zeros elsewhere. The six off-diagonal $E_{ij}$ with $i\ne j$ each lie in ${\mathfrak{sl}}_3\mathbb{C}$.

Step 3. Read off the action of $H_1$ on $E_{12}$. Compute $[H_1,E_{12}]=H_1{E}_{12}-E_{12}{H}_1=1\cdot E_{12}-(-1)\cdot E_{12}=2E_{12}$. Similarly $[H_2,E_{12}]=0\cdot E_{12}-1\cdot E_{12}=-E_{12}$. So $E_{12}$ is a weight vector with weight $(2,-1)$ — this is the simple root ${\alpha }_1$ in coordinates.

Step 4. Repeat for all six off-diagonal directions. The weights come out as ${\alpha }_1=(2,-1)$ for $E_{12}$, ${\alpha }_2=(-1,2)$ for $E_{23}$, ${\alpha }_1+{\alpha }_2=(1,1)$ for $E_{13}$, and the negatives $-{\alpha }_1,-{\alpha }_2,-({\alpha }_1+{\alpha }_2)$ for $E_{21},E_{32},E_{31}$. The six weights form a regular hexagon centred at the origin.

Step 5. The remaining two dimensions are the Cartan subalgebra itself. Both $H_1$ and $H_2$ have weight $(0,0)$: each commutes with itself. So the zero-weight space has dimension two, and the total weight-multiplicity count is $6\times 1+1\times 2=8=\dim {\mathfrak{sl}}_3\mathbb{C}$.

What this shows: the adjoint representation is exactly the hexagonal weight diagram, with six outer dots of multiplicity one and one inner dot of multiplicity two at the centre. The integer parameters $(a,b)=(1,1)$ identify this irreducible as $V_{1,1}$, and the Weyl dimension formula confirms $\dim V_{1,1}=(1+1)(1+1)(1+1+2)/2=8$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The Lie algebra ${\mathfrak{sl}}_3\mathbb{C}$ is the eight-dimensional complex Lie algebra of traceless $3\times 3$ complex matrices under the commutator bracket. A standard basis adapted to the root-space decomposition consists of two diagonal Cartan generators $$ H_1 = \begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 0 \end{pmatrix}, \qquad H_2 = \begin{pmatrix} 0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \end{pmatrix}, $$ and the six off-diagonal matrix units $E_{ij}$ for $i\ne j$, where $E_{ij}$ has a $1$ in the $(i,j)$ entry and zeros elsewhere. The bracket relations are determined by $[E_{ij},E_{k\ell }]={\delta }_{jk}{E}_{i\ell }-{\delta }_{\ell i}{E}_{kj}$ together with the diagonal-bracket relations $[H_a,E_{ij}]=({\delta }_{ai}-{\delta }_{aj}-{\delta }_{a,i-1}+{\delta }_{a,j-1})E_{ij}$ for the chosen Cartan basis. Equivalently, writing $L_i\in {\mathfrak{h}}^{\ast }$ for the linear functional reading off the $i$-th diagonal entry of a diagonal matrix, the off-diagonal $E_{ij}$ is a weight vector of weight $L_i-L_j$ on the Cartan subalgebra $\mathfrak{h}$.

The Cartan subalgebra $\mathfrak{h}\subseteq {\mathfrak{sl}}_3\mathbb{C}$ is the two-dimensional abelian subalgebra of diagonal traceless matrices, spanned by $H_1$ and $H_2$ (or equivalently by any two of $L_1-L_2$, $L_2-L_3$, $L_1-L_3$). The root system $\Phi \subseteq {\mathfrak{h}}^{\ast }$ is the set of non-zero weights of the adjoint action of $\mathfrak{h}$ on ${\mathfrak{sl}}_3\mathbb{C}$: $$ \Phi = { \pm(L_1 - L_2), \pm(L_2 - L_3), \pm(L_1 - L_3) }, $$ six vectors arranged hexagonally in the two-dimensional space ${\mathfrak{h}}_{\mathbb{R}}^{\ast }$. The simple roots are ${\alpha }_1=L_1-L_2$ and ${\alpha }_2=L_2-L_3$, and the positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$. The Cartan matrix with respect to this base is $$ A = \begin{pmatrix} 2 & -1 \ -1 & 2 \end{pmatrix}, $$ where $A_{ij}={\alpha }_i(H_{{\alpha }_j})=2({\alpha }_i,{\alpha }_j)/({\alpha }_j,{\alpha }_j)$ and the inner product on ${\mathfrak{h}}^{\ast }$ is normalised so that simple roots have squared length $2$.

A representation of ${\mathfrak{sl}}_3\mathbb{C}$ on a complex vector space $V$ is a Lie algebra homomorphism $\rho :{\mathfrak{sl}}_3\mathbb{C}\to \mathfrak{gl}(V)$. A weight vector of weight $\mu \in {\mathfrak{h}}^{\ast }$ is a non-zero $v\in V$ satisfying $\rho (H)v=\mu (H)v$ for all $H\in \mathfrak{h}$. The weight space $V_{\mu }$ is the simultaneous $\mathfrak{h}$-eigenspace at weight $\mu$. Because the off-diagonal generators shift weights — $\rho (E_{ij})V_{\mu }\subseteq V_{\mu +L_i-L_j}$ by the same calculation as in the ${\mathfrak{sl}}_2$ case — every finite-dimensional representation decomposes as $V={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{V}_{\mu }$.

A highest-weight vector is a weight vector $v_{\lambda }$ annihilated by all the positive root operators $\rho (E_{12}),\rho (E_{23}),\rho (E_{13})$. The fundamental weights ${\lambda }_1,{\lambda }_2\in {\mathfrak{h}}^{\ast }$ are defined by duality with the simple coroots: ${\lambda }_i(H_{{\alpha }_j})={\delta }_{ij}$. Concretely, ${\lambda }_1=L_1$ and ${\lambda }_2=L_1+L_2=-L_3$ when restricted to the ${\mathfrak{h}}^{\ast }$ subject to $L_1+L_2+L_3=0$. A weight $\lambda$ is integral dominant when $\lambda =a{\lambda }_1+b{\lambda }_2$ with $a,b\in {\mathbb{Z}}_{\ge 0}$. Writing $V_{a,b}$ for the irreducible representation with highest weight $a{\lambda }_1+b{\lambda }_2$ unties the parametrisation: $V_{1,0}=V$ is the defining representation on ${\mathbb{C}}^3$ with highest weight $L_1={\lambda }_1$, and $V_{0,1}=V^{\ast }$ is its dual with highest weight $-L_3={\lambda }_2$.

Counterexamples to common slips

• The Cartan subalgebra is two-dimensional, not three. Three diagonal matrices $\mathrm{diag}(a_1,a_2,a_3)$ span a three-dimensional space, but the trace constraint $a_1+a_2+a_3=0$ cuts one dimension. The two generators $H_1,H_2$ chosen above form a basis of the resulting $2$-plane.
• The Weyl group of $A_2$ is the symmetric group $S_3$, acting on the three weights $L_1,L_2,L_3$ by permutation, of order $6$ — not $\mathbb{Z}/2$ as in the rank-one case. The hexagonal weight diagram has the dihedral symmetry of the regular hexagon, generated by the two simple reflections $s_{{\alpha }_1},s_{{\alpha }_2}$.
• The two fundamental representations $V$ and $V^{\ast }$ are not isomorphic — they have opposite highest weights ${\lambda }_1$ versus ${\lambda }_2$ and their weight diagrams are mirror images. For ${\mathfrak{sl}}_2$, by contrast, $V\cong V^{\ast }$ because there is one fundamental and the Weyl group identifies a weight with its negative.

Key theorem with proof [Intermediate+]

Theorem (Classification of finite-dimensional irreducible representations of ${\mathfrak{sl}}_3\mathbb{C}$). The finite-dimensional irreducible representations of ${\mathfrak{sl}}_3\mathbb{C}$ are in bijection with pairs of non-negative integers $(a,b)\in {\mathbb{Z}}_{\ge 0}^2$. For every such pair there exists a finite-dimensional irreducible representation $V_{a,b}$ of ${\mathfrak{sl}}_3\mathbb{C}$, unique up to isomorphism, characterised by the existence of a highest-weight vector $v_{a,b}\in V_{a,b}$ annihilated by $E_{12},E_{23},E_{13}$ and satisfying $H_1{v}_{a,b}=a{v}_{a,b}$, $H_2{v}_{a,b}=b{v}_{a,b}$. Conversely, every finite-dimensional irreducible representation of ${\mathfrak{sl}}_3\mathbb{C}$ is isomorphic to some $V_{a,b}$.

Proof. The proof has three parts: existence (construct $V_{a,b}$), uniqueness (any two irreducibles with the same highest weight are isomorphic), and integrality (the highest weight must be integral dominant).

Part 1: integrality from existence of a highest-weight vector. Let $W$ be a finite-dimensional irreducible representation. The Cartan subalgebra $\mathfrak{h}$ acts by commuting diagonalisable operators on $W$ (by ${\mathfrak{sl}}_2$-triple arguments applied to each $H_{{\alpha }_i}$; each acts diagonalisably because $W$ restricts to a finite-dimensional representation of the corresponding ${\mathfrak{sl}}_2$, which is completely reducible into the $V_n$ each diagonalising $H$). So $W$ decomposes as $W={⨁}_{\mu }{W}_{\mu }$ into $\mathfrak{h}$-weight spaces, and the positive root operators $E_{12},E_{23},E_{13}$ shift weight by ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$ respectively.

Choose a partial order on weights: $\mu ⪰\nu$ when $\mu -\nu$ is a non-negative integer combination of positive roots. Finite-dimensionality of $W$ together with the weight-shift property implies the set of weights of $W$ has a maximal element $\lambda$ with respect to $⪰$. Any weight vector $v_{\lambda }\in W_{\lambda }$ is annihilated by $E_{12},E_{23},E_{13}$ (since $\lambda +{\alpha }_i$ is strictly greater than $\lambda$, but $\lambda$ is maximal, so the corresponding image is zero). So $v_{\lambda }$ is a highest-weight vector.

Restrict the ${\mathfrak{sl}}_2$-triple $(H_{{\alpha }_1},E_{12},E_{21})$ to act on $W$. By the ${\mathfrak{sl}}_2$ classification (see 07.06.11), the chain of vectors $v_{\lambda },E_{21}{v}_{\lambda },E_{21}^2{v}_{\lambda },\dots$ generates a finite-dimensional ${\mathfrak{sl}}_2$-submodule that decomposes into irreducibles $V_n$, each with integer top eigenvalue $n$. The top eigenvalue on the chain through $v_{\lambda }$ is $\lambda (H_{{\alpha }_1})=a$, so $a\in {\mathbb{Z}}_{\ge 0}$. The same argument with the ${\mathfrak{sl}}_2$-triple $(H_{{\alpha }_2},E_{23},E_{32})$ forces $b=\lambda (H_{{\alpha }_2})\in {\mathbb{Z}}_{\ge 0}$. So $\lambda =a{\lambda }_1+b{\lambda }_2$ is integral dominant.

Part 2: existence. Given $(a,b)\in {\mathbb{Z}}_{\ge 0}^2$, construct $V_{a,b}$ as the irreducible quotient of the Verma module $M(a,b)=U({\mathfrak{sl}}_3\mathbb{C}){\otimes }_{U(\mathfrak{b})}{\mathbb{C}}_{a{\lambda }_1+b{\lambda }_2}$, where $\mathfrak{b}$ is the Borel subalgebra spanned by $\mathfrak{h}$ and the positive root operators, and ${\mathbb{C}}_{\lambda }$ is the one-dimensional $\mathfrak{b}$-module on which $\mathfrak{h}$ acts by $\lambda$ and the positive root operators act by zero. The Verma module is infinite-dimensional but admits a unique maximal proper submodule $N(a,b)$, and the quotient $L(a,b)=M(a,b)/N(a,b)$ is the irreducible highest-weight module of weight $a{\lambda }_1+b{\lambda }_2$.

The argument that $L(a,b)$ is finite-dimensional uses the integral-dominance condition $a,b\in {\mathbb{Z}}_{\ge 0}$ together with the structure of the Weyl group. By the Harish-Chandra/Verma analysis (see 07.06.06), the submodule generated by $F_{{\alpha }_1}^{a+1}{v}_{\lambda }$ and $F_{{\alpha }_2}^{b+1}{v}_{\lambda }$ (where $F_{{\alpha }_i}=E_{i+1,i}$) is the kernel making $L(a,b)$ finite-dimensional, with weight set equal to the Weyl-group orbit of weights $\mu$ with $\mu ⪯a{\lambda }_1+b{\lambda }_2$ and $\mu$ in the convex hull of the Weyl-orbit of $a{\lambda }_1+b{\lambda }_2$. Setting $V_{a,b}=L(a,b)$ completes the existence step.

Part 3: uniqueness. Suppose $W$ and $W^{\prime }$ are two finite-dimensional irreducibles of ${\mathfrak{sl}}_3\mathbb{C}$ with the same highest weight $\lambda =a{\lambda }_1+b{\lambda }_2$. Pick highest-weight vectors $v\in W$ and $v^{\prime }\in W^{\prime }$. The tensor product $W\otimes W^{\prime \ast }$ contains a ${\mathfrak{sl}}_3\mathbb{C}$-invariant element by the matrix-coefficient $\varphi (w)=⟨v^{\prime },{\pi }^{\prime }(g)v^{\prime }⟩$ where $g\in U({\mathfrak{sl}}_3\mathbb{C})$ acts to take $v$ to $w$; equivalently, by Schur's lemma the space ${\mathrm{Hom}}_{{\mathfrak{sl}}_3}(W,W^{\prime })$ is one-dimensional, generated by the unique morphism sending $v$ to $v^{\prime }$. This morphism is non-zero, hence an isomorphism by irreducibility. So $W\cong W^{\prime }$, completing the uniqueness step. $\square$

Theorem (Weyl dimension formula for ${\mathfrak{sl}}_3\mathbb{C}$). The dimension of the irreducible representation $V_{a,b}$ is $$ \dim V_{a, b} = \tfrac{1}{2} (a + 1)(b + 1)(a + b + 2). $$

Proof. Apply the Weyl dimension formula (see 07.06.08): $$ \dim V_\lambda = \prod_{\alpha \in \Phi^+} \frac{(\lambda + \rho, \alpha)}{(\rho, \alpha)}, $$ where $\rho ={\lambda }_1+{\lambda }_2=(1,1)$ in fundamental-weight coordinates is the half-sum of positive roots. The three positive roots of $A_2$ are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$. In coordinates where the inner product is the standard Euclidean one rescaled so $({\alpha }_i,{\alpha }_i)=2$:

• $(\rho ,{\alpha }_1)=1$, $(\rho ,{\alpha }_2)=1$, $(\rho ,{\alpha }_1+{\alpha }_2)=2$;
• $(\lambda +\rho ,{\alpha }_1)=a+1$, $(\lambda +\rho ,{\alpha }_2)=b+1$, $(\lambda +\rho ,{\alpha }_1+{\alpha }_2)=a+b+2$.

Multiplying the three ratios: $\dim V_{a,b}=\frac{(a+1)(b+1)(a+b+2)}{1\cdot 1\cdot 2}$, as claimed. $\square$

Bridge. This dimension formula builds toward the general Weyl character formula and identifies the integer-pair classification with explicit polynomial growth in $(a,b)$. The foundational reason the highest-weight theorem works in rank two is exactly the rank-one ladder structure applied along each of the two simple roots: the integrality $a,b\in {\mathbb{Z}}_{\ge 0}$ is the same integrality that forced $n\in {\mathbb{Z}}_{\ge 0}$ in the ${\mathfrak{sl}}_2$ case (see 07.06.11), now propagated along two independent simple-root directions instead of one. This is exactly the central insight that generalises to all of Cartan-Killing: every irreducible representation of a semisimple Lie algebra is determined by an integral dominant highest weight, and the dimension is computed by the Weyl product formula over positive roots. The hexagonal weight diagram of $V_{a,b}$ is dual to the integer-pair parametrisation through the explicit map from a dominant weight to the convex hull of its Weyl-orbit, and putting these together identifies $V_{a,b}$ with the data $(a,b)$ in a Weyl-symmetric and combinatorially explicit way. The bridge is that the two fundamental representations $V$ and $V^{\ast }$ of ${\mathfrak{sl}}_3$ generalise the single fundamental representation of ${\mathfrak{sl}}_2$ to a rank-two pattern, and the Weyl dimension formula appears again in 07.06.08 (Weyl dimension formula) as the universal answer that specialises here to $(a+1)(b+1)(a+b+2)/2$.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib has the abstract Lie-algebra infrastructure in Mathlib.Algebra.Lie.Basic, the matrix-bracket construction in Mathlib.Algebra.Lie.OfAssociative, and weight-module scaffolding in Mathlib.Algebra.Lie.Weights.Basic, but no named development of the representation theory of ${\mathfrak{sl}}_3\mathbb{C}$. The intended formalisation reads schematically:

import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Trace

namespace Codex.Sl3

/-- The Lie algebra sl₃ ℂ as traceless 3×3 complex matrices. -/
abbrev sl3 : Type := { M : Matrix (Fin 3) (Fin 3) ℂ // M.trace = 0 }

/-- Cartan generator H₁ = diag(1, -1, 0). -/
noncomputable def H1 : sl3 := sorry

/-- Cartan generator H₂ = diag(0, 1, -1). -/
noncomputable def H2 : sl3 := sorry

/-- The Cartan subalgebra is spanned by H₁ and H₂; bracket vanishes. -/
theorem cartan_commute : ⁅H1, H2⁆ = 0 := sorry

/-- Cartan matrix of A₂. -/
noncomputable def cartanMatrix : Matrix (Fin 2) (Fin 2) ℤ :=
  ![![2, -1], ![-1, 2]]

/-- The irreducible representation V_{a,b} with highest weight a λ₁ + b λ₂. -/
noncomputable def V (a b : ℕ) : Type := sorry

/-- Weyl dimension formula for sl₃. -/
theorem weylDim (a b : ℕ) :
    Module.finrank ℂ (V a b) = (a + 1) * (b + 1) * (a + b + 2) / 2 :=
  sorry

/-- Highest-weight classification: every finite-dimensional irreducible
    representation of sl₃ ℂ is V_{a,b} for some unique (a, b). -/
theorem highestWeight_classification
    {W : Type*} [AddCommGroup W] [Module ℂ W] [Finite W]
    [LieModule ℂ sl3 W] (irred : IsIrreducible ℂ sl3 W) :
    ∃ a b : ℕ, Nonempty (W ≃ₗ⁅sl3⁆ V a b) :=
  sorry

/-- The adjoint representation is V_{1,1}. -/
theorem adjoint_iso_V11 :
    Nonempty (sl3 ≃ₗ⁅sl3⁆ V 1 1) :=
  sorry

end Codex.Sl3

The proof gap is twofold: first, the linear-algebra computations of the bracket relations and the Cartan-subalgebra decomposition are finite-rank matrix manipulations that Mathlib's matrix infrastructure supports, but the named theorems identifying the Cartan-subalgebra-with-diagonal-traceless and the six root spaces with the off-diagonal matrix units $E_{ij}$ are not packaged. Second, the highest-weight classification reduces by a Verma-module quotient argument to the integrality $a,b\in {\mathbb{Z}}_{\ge 0}$, where the rank-one ladder applied along each simple-root direction propagates the integrality from the ${\mathfrak{sl}}_2$ case (07.06.11); Mathlib has the abstract weight-module scaffolding but not the concrete $A_2$ implementation. The Weyl dimension formula in this case reduces to evaluating a product of three rational expressions, formalisable as a numerical lemma but not currently packaged. The adjoint-representation identification $V_{1,1}$ would follow from a weight-multiplicity count.

Advanced results [Master]

Theorem (Convex-hull description of weights). Let $V_{a,b}$ be the irreducible representation of ${\mathfrak{sl}}_3\mathbb{C}$ with highest weight $\lambda =a{\lambda }_1+b{\lambda }_2$. The set of weights of $V_{a,b}$ is the intersection of the Weyl-group orbit $W\cdot \lambda$ with the convex hull $\mathrm{conv}(W \cdot \lambda) \subset \mathfrak{h}^{\mathbb{R}}$,togetherwithallweights$\mu$inthisconvexhullsatisfying$\mu - \lambda \in \mathbb{Z} \Phi$ (the root lattice). The multiplicities are determined by the Freudenthal recursion* $$ ((\lambda + \rho, \lambda + \rho) - (\mu + \rho, \mu + \rho)) m\mu = 2 \sum_{\alpha \in \Phi^+} \sum_{k \geq 1} m_{\mu + k\alpha} (\mu + k\alpha, \alpha). $$ The outermost weights (those on the boundary of the convex hull, in the Weyl-orbit of $\lambda$) have multiplicity exactly $1$, and each interior shell increases multiplicities by $1$ until the central shell is reached.

This is the rank-two version of the convex-hull description that already appears for ${\mathfrak{sl}}_2$ as a line segment $[-n,n]$ with multiplicity $1$ at each integer point. The hexagonal weight diagram is the Weyl orbit of the highest weight, and the shells from the boundary inward have multiplicities $1,2,3,\dots$ stabilising once a shell becomes triangular (in pure $V_{a,0}$ or $V_{0,b}$ cases the diagram is a single triangle and all multiplicities are $1$).

Theorem (Clebsch-Gordan decomposition for fundamental representations). The tensor products of the two fundamental representations of ${\mathfrak{sl}}_3\mathbb{C}$ decompose as $$ V \otimes V \cong V_{2, 0} \oplus V_{0, 1} = \mathrm{Sym}^2(V) \oplus \Lambda^2(V), $$ $$ V \otimes V^* \cong V_{1, 1} \oplus V_{0, 0} = \mathrm{Ad}(\mathfrak{sl}3) \oplus \mathbb{C}, $$ $$ V^* \otimes V^* \cong V{0, 2} \oplus V_{1, 0} = \mathrm{Sym}^2(V^) \oplus \Lambda^2(V^). $$ The first and third decompositions express the splitting of a $9$-dimensional tensor product into symmetric and antisymmetric parts; the middle decomposition identifies the adjoint $V_{1,1}$ as the traceless part of $V \otimes V^ \cong \mathrm{End}(V)$.*

This is the rank-two analogue of the ${\mathfrak{sl}}_2$ Clebsch-Gordan rule $V_m\otimes V_n={⨁}_k{V}_{m+n-2k}$. The middle identity $V\otimes V^{\ast }=V_{1,1}\oplus \mathbb{C}$ is the canonical Lie-algebraic statement that the endomorphism algebra of $V$ decomposes as the trace-zero subspace (the adjoint) plus the scalar subspace (the identity). In physics this is the SU(3) tensor decomposition $3\otimes \bar{3}=8\oplus 1$ — the eight-fold way of Gell-Mann and Ne'eman, where the eight is the adjoint and the one is the trace.

Theorem (Steinberg / Littlewood-Richardson for ${\mathfrak{sl}}_3$). The Clebsch-Gordan multiplicities for $V_{a_1,b_1}\otimes V_{a_2,b_2}$ are given by the rank-two specialisation of the Littlewood-Richardson rule, equivalently by Steinberg's character formula: $$ \chi_{V_{a_1, b_1}} \chi_{V_{a_2, b_2}} = \sum_{(a, b)} N^{(a, b)}{(a_1, b_1), (a_2, b_2)} \chi{V_{a, b}}, $$ where $N_{(a_1,b_1),(a_2,b_2)}^{(a,b)}$ counts semistandard Young tableaux of shape $(a+b,b)$ (in the two-row Young-diagram language for ${\mathfrak{sl}}_3$) with content determined by the input weights, modulo a Weyl-group anti-symmetrisation.

The general ${\mathfrak{sl}}_3$ Clebsch-Gordan rule is not as compact as the ${\mathfrak{sl}}_2$ formula but is still computable in polynomial time and gives the full structure of the representation ring of ${\mathfrak{sl}}_3\mathbb{C}$. The rule appears again in 07.06.07 (Weyl character formula) as a character-theoretic identity.

Theorem (Casimir element and central characters). The quadratic Casimir element $$ C = \tfrac{1}{2} H_1^2 + \tfrac{1}{2} H_2^2 + \tfrac{1}{2} H_1 H_2 + \tfrac{1}{2} H_2 H_1 + \sum_{\alpha \in \Phi^+} (E_\alpha F_\alpha + F_\alpha E_\alpha) $$ lies in the centre of the universal enveloping algebra $U({\mathfrak{sl}}_3\mathbb{C})$. On the irreducible representation $V_{a,b}$ it acts as the scalar $$ C |{V{a, b}} = (\lambda + \rho, \lambda + \rho) - (\rho, \rho) = \tfrac{1}{3}(a^2 + b^2 + ab) + a + b, $$ where $\lambda =a{\lambda }_1+b{\lambda }_2$ and $\rho ={\lambda }_1+{\lambda }_2$.

The Casimir scalar in rank two is no longer a single integer parameter but a quadratic polynomial in $(a,b)$. The Harish-Chandra centre $Z(U({\mathfrak{sl}}_3\mathbb{C}))$ is a polynomial algebra in two generators (the quadratic and cubic Casimirs), reflecting the rank-two parametrisation; this is the rank-two instance of the general Harish-Chandra isomorphism $Z(U(\mathfrak{g}))\cong S(\mathfrak{h}{)}^W$.

Theorem (Borel-Weil realisation for ${\mathfrak{sl}}_3$). Every irreducible $V_{a,b}$ of ${\mathfrak{sl}}_3\mathbb{C}$ is realised geometrically as $H^0({\mathrm{SL}}_3\mathbb{C}/B,{\mathcal{L}}_{a,b})$, the space of holomorphic global sections of the line bundle ${\mathcal{L}}_{a,b}$ on the flag variety ${\mathrm{SL}}_3\mathbb{C}/B={\mathrm{Fl}}_3(\mathbb{C})$ (the full flag variety of ${\mathbb{C}}^3$) twisted by the character $(a,b)$ of the maximal torus.

This is the geometric counterpart to the algebraic classification, and the rank-two case of the Borel-Weil theorem (see 07.06.09). The flag variety ${\mathrm{Fl}}_3(\mathbb{C})$ is a six-dimensional complex projective variety, the simplest case of a complete flag variety beyond $\mathbb{C}{P}^1={\mathrm{Fl}}_2(\mathbb{C})$, and the line bundles ${\mathcal{L}}_{a,b}$ are indexed by pairs of non-negative integers exactly as the irreducible representations are.

Synthesis. The representation theory of ${\mathfrak{sl}}_3\mathbb{C}$ is the foundational reason that the rank-two case of Cartan-Killing classification works, and it is exactly the rank-one ${\mathfrak{sl}}_2$ ladder structure (07.06.11) propagated along two independent simple-root directions. The central insight is that every irreducible representation $V_{a,b}$ is generated by a highest-weight vector annihilated by both positive simple-root operators, with the two independent integers $(a,b)$ replacing the single integer $n$ of ${\mathfrak{sl}}_2$; this is exactly the local model for every weight chain in every higher-rank Lie algebra, with the rank two simply tracking two simultaneously available raising-lowering directions instead of one. The hexagonal weight diagram with its $S_3$ Weyl symmetry and shell-by-shell multiplicity structure builds toward the general weight-multiplicity machinery of arbitrary semisimple Lie algebras, with the Weyl-orbit-and-convex-hull description appearing again in 07.06.07 (Weyl character formula) as the universal recipe. The Weyl dimension formula $(a+1)(b+1)(a+b+2)/2$ generalises the rank-one $\dim V_n=n+1$ by adding a quadratic factor for the third positive root ${\alpha }_1+{\alpha }_2$.

Putting these together, the pattern is clear: ${\mathfrak{sl}}_{n+1}$ has $n$ fundamental representations (the exterior powers ${\Lambda }^{k}V$), a Cartan subalgebra of dimension $n$, a root system $A_n$ with $n(n+1)$ roots, a Weyl group $S_{n+1}$, and a Weyl dimension formula given by an $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$-fold product over positive roots — and every step of this pattern is the rank-$n$ propagation of the rank-one engine. The bridge between ${\mathfrak{sl}}_3$ and the geometric / topological side of representation theory is the Borel-Weil theorem 07.06.09: $V_{a,b}$ is geometrically the space of holomorphic sections of the line bundle ${\mathcal{L}}_{a,b}$ on the flag variety ${\mathrm{SL}}_3\mathbb{C}/B={\mathrm{Fl}}_3(\mathbb{C})$, identifying algebraic highest-weight data with cohomological data on a complex projective variety.

The duality between the algebraic and the combinatorial picture sharpens at rank two. The Freudenthal multiplicity formula computes the integer multiplicity of every weight in $V_{a,b}$ recursively from the highest weight inward, with the shells of the hexagonal diagram carrying multiplicities $1,2,3,\dots$ in stair-step fashion until the central shell is reached. The Weyl-orbit-and-convex-hull description identifies the weight set with the lattice points of the convex polygon $\mathrm{conv}(S_3\cdot (a{\lambda }_1+b{\lambda }_2))$. Steinberg's general Clebsch-Gordan formula gives the tensor-product decomposition $V_{a_1,b_1}\otimes V_{a_2,b_2}$ via inclusion-exclusion over $S_3$, generalising the Pieri rule and culminating in the full Littlewood-Richardson description for ${\mathfrak{sl}}_3$ via two-row Young tableaux. Quantum deformation produces $U_q({\mathfrak{sl}}_3)$ with the same Cartan-and-root-data, with crystal bases (Kashiwara 1991) decorating the hexagonal weight diagram by Kashiwara operators that move along the six root directions; the classical limit $q\to 1$ recovers the ${\mathfrak{sl}}_3$ representation theory of this unit. Affine deformation produces $\widehat{{\mathfrak{sl}}_3}$ with infinite-dimensional level-and-weight integrable representations, and the extended Dynkin diagram for $\widehat{{\mathfrak{sl}}_3}$ adds a third node forming a triangle, recapitulating the rank-two hexagonal symmetry at the affine level. The bridge is that the hexagonal weight diagram of ${\mathfrak{sl}}_3$ is the local model for every two-dimensional weight projection in every higher-rank semisimple Lie algebra, just as the ${\mathfrak{sl}}_2$ ladder is the local model for every one-dimensional weight chain.

Full proof set [Master]

Theorem (classification of irreducible ${\mathfrak{sl}}_3\mathbb{C}$-representations), proof. Given in full in the Intermediate-tier section: integrality $a,b\in {\mathbb{Z}}_{\ge 0}$ follows from applying the ${\mathfrak{sl}}_2$-classification (07.06.11) to each of the two ${\mathfrak{sl}}_2$-triples $(H_{{\alpha }_i},E_{{\alpha }_i},F_{{\alpha }_i})$ for $i=1,2$. Existence of $V_{a,b}$ as the irreducible quotient of the Verma module $M(a,b)$ uses the integrality to ensure finite-dimensionality. Uniqueness follows from Schur's lemma applied to the one-dimensional space of highest-weight maps between two irreducibles with the same highest weight. $\square$

Proposition (root-space decomposition of ${\mathfrak{sl}}_3\mathbb{C}$). Under the adjoint action of the Cartan subalgebra $\mathfrak{h}$ spanned by $H_1,H_2$, the Lie algebra ${\mathfrak{sl}}_3\mathbb{C}$ decomposes as ${\mathfrak{sl}}_3\mathbb{C}=\mathfrak{h}\oplus {⨁}_{\alpha \in \Phi }{\mathfrak{g}}_{\alpha }$ where $\Phi =\{\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2)\}$ and each root space ${\mathfrak{g}}_{\alpha }$ is one-dimensional, spanned by the appropriate $E_{ij}$.

Proof. The Cartan subalgebra $\mathfrak{h}$ has dimension $2$, spanned by $H_1=\mathrm{diag}(1,-1,0)$ and $H_2=\mathrm{diag}(0,1,-1)$. The total dimension of ${\mathfrak{sl}}_3\mathbb{C}$ is $3^2-1=8$, leaving $6$ dimensions for the root spaces.

Compute the action of $H_1$ and $H_2$ on each off-diagonal matrix unit. For $E_{ij}$ with $i\ne j$: $$ [H_1, E_{ij}] = (\delta_{1i} - \delta_{1j} + \delta_{2j} - \delta_{2i}) E_{ij}, $$ $$ [H_2, E_{ij}] = (\delta_{2i} - \delta_{2j} + \delta_{3j} - \delta_{3i}) E_{ij}. $$ A short check: $[H_1,E_{12}]=(1-0+1-0)E_{12}=2E_{12}$ (since ${\delta }_{2j}={\delta }_{22}=1$, ${\delta }_{1i}={\delta }_{11}=1$), and similarly all six $E_{ij}$ for $i\ne j$ are simultaneous $H_1,H_2$ eigenvectors. The resulting weights are $$ E_{12} \mapsto (2, -1) = \alpha_1, \quad E_{23} \mapsto (-1, 2) = \alpha_2, \quad E_{13} \mapsto (1, 1) = \alpha_1 + \alpha_2, $$ $$ E_{21} \mapsto (-2, 1) = -\alpha_1, \quad E_{32} \mapsto (1, -2) = -\alpha_2, \quad E_{31} \mapsto (-1, -1) = -\alpha_1 - \alpha_2. $$ The six off-diagonal $E_{ij}$ each span a distinct one-dimensional weight space, completing the decomposition. $\square$

Proposition (Casimir centrality for ${\mathfrak{sl}}_3$). The quadratic Casimir element $C={\sum }_{a}B(X_a,X_a{)}^{-1}{X}_a^2$, computed with respect to the Killing form $B$ on a basis $\{X_a\}$ of ${\mathfrak{sl}}_3\mathbb{C}$, lies in the centre of the universal enveloping algebra $U({\mathfrak{sl}}_3\mathbb{C})$.

Proof. The Killing form $B(X,Y)=\mathrm{tr}({\mathrm{ad}}_X{\mathrm{ad}}_Y)$ on ${\mathfrak{sl}}_3\mathbb{C}$ restricts to $\mathfrak{h}\times \mathfrak{h}$ as a non-degenerate bilinear form (the diagonal-traceless form scaled by a factor of $2(n+1)=8$ for $n+1=3$). On root spaces ${\mathfrak{g}}_{\alpha }\times {\mathfrak{g}}_{-\alpha }$ it is non-degenerate, and it pairs ${\mathfrak{g}}_{\alpha }$ with ${\mathfrak{g}}_{\beta }$ to zero when $\alpha +\beta \ne 0$ (by the invariance $B([X,Y],Z)=B(X,[Y,Z])$ together with the weight argument).

The centrality $[Y,C]=0$ for all $Y\in {\mathfrak{sl}}_3\mathbb{C}$ follows from the general identity for the quadratic Casimir of any Lie algebra equipped with a non-degenerate invariant bilinear form: choose a basis $\{X_a\}$ with dual basis $\{X^a\}$ such that $B(X_a,X^b)={\delta }_a^b$, define $C={\sum }_a{X}_a{X}^a$, and compute $[Y,C]={\sum }_a[Y,X_a]X^a+{\sum }_a{X}_a[Y,X^a]={\sum }_{a,b}B([Y,X_a],X^b)X_b{X}^a+{\sum }_{a,b}{X}_{a}B([Y,X^b],X_b{)}^{-1}{X}^a\cdot {\delta }_b^a$. A telescoping argument using the symmetry $B([Y,X_a],X_b)=-B(X_a,[Y,X_b])$ (the invariance of $B$) cancels every term in pairs, giving $[Y,C]=0$. The argument applies in particular to ${\mathfrak{sl}}_3\mathbb{C}$, identifying the explicit quadratic Casimir of the theorem above as central. The scalar eigenvalue on $V_{a,b}$ is computed by evaluating $C$ on the highest-weight vector and using $E_{\alpha }{v}_{\lambda }=0$ for positive roots. $\square$

Proposition (Weyl dimension formula, ${\mathfrak{sl}}_3$ case). The dimension of $V_{a,b}$ is $(a+1)(b+1)(a+b+2)/2$.

Proof. The general Weyl dimension formula (see 07.06.08) reads $\dim V_{\lambda }={\prod }_{\alpha \in {\Phi }^{+}}(\lambda +\rho ,\alpha )/(\rho ,\alpha )$ where $\rho$ is the half-sum of positive roots. For ${\mathfrak{sl}}_3$ with positive roots ${\Phi }^{+}=\{{\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2\}$ and $\rho ={\alpha }_1+{\alpha }_2={\lambda }_1+{\lambda }_2$ (using the standard identity that $\rho$ equals the sum of fundamental weights): $$ (\rho, \alpha_1) = (\lambda_1 + \lambda_2, \alpha_1) = (\lambda_1, \alpha_1) + (\lambda_2, \alpha_1) = 1 + 0 = 1, $$ using $({\lambda }_i,{\alpha }_j)={\delta }_{ij}({\alpha }_j,{\alpha }_j)/2={\delta }_{ij}$ in the rescaled normalisation $({\alpha }_i,{\alpha }_i)=2$. Similarly $(\rho ,{\alpha }_2)=1$ and $(\rho ,{\alpha }_1+{\alpha }_2)=2$. For $\lambda +\rho =(a+1){\lambda }_1+(b+1){\lambda }_2$: $$ (\lambda + \rho, \alpha_1) = a + 1, \quad (\lambda + \rho, \alpha_2) = b + 1, \quad (\lambda + \rho, \alpha_1 + \alpha_2) = a + b + 2. $$ The product gives $\dim V_{a,b}=(a+1)(b+1)(a+b+2)/(1\cdot 1\cdot 2)=(a+1)(b+1)(a+b+2)/2$. $\square$

Proposition (Clebsch-Gordan $V \otimes V^ \cong V_{1, 1} \oplus \mathbb{C}$).* The tensor product of the standard and dual representations decomposes as $V \otimes V^ \cong V_{1, 1} \oplus V_{0, 0}$.*

Proof. Dimension check: $\dim (V\otimes V^{\ast })=3\cdot 3=9=8+1=\dim V_{1,1}+\dim V_{0,0}$.

Identify the zero-weight component. The natural pairing $V\otimes V^{\ast }\to \mathbb{C}$ given by $v\otimes \varphi \mapsto \varphi (v)$ is a ${\mathfrak{sl}}_3\mathbb{C}$-equivariant map, with kernel of dimension $9-1=8$. This kernel is the space of traceless tensors, i.e., ${\mathrm{End}}_0(V)=\{T\in \mathrm{End}(V):\mathrm{tr}(T)=0\}$, which is exactly ${\mathfrak{sl}}_3\mathbb{C}$. The action of ${\mathfrak{sl}}_3\mathbb{C}$ on ${\mathrm{End}}_0(V)$ by $X\cdot T=XT-TX$ is the adjoint action, which is the irreducible representation $V_{1,1}$ by the root-space decomposition argument.

The identification of the adjoint representation as $V_{1,1}$: the highest weight of the adjoint representation is the highest root ${\alpha }_1+{\alpha }_2={\lambda }_1+{\lambda }_2$, exactly the highest weight of $V_{1,1}$ corresponding to $(a,b)=(1,1)$. So $V\otimes V^{\ast }\cong V_{1,1}\oplus V_{0,0}$ as claimed. $\square$

Connections [Master]

• Representations of ${\mathfrak{sl}}_2\mathbb{C}$ 07.06.11. The rank-one case is the direct prerequisite and the technical engine: the integrality $a,b\in {\mathbb{Z}}_{\ge 0}$ of the highest-weight classification for ${\mathfrak{sl}}_3$ follows from applying the ${\mathfrak{sl}}_2$ classification along each of the two ${\mathfrak{sl}}_2$-triples $(H_{{\alpha }_i},E_{{\alpha }_i},F_{{\alpha }_i})$ for $i=1,2$. Every weight chain in ${\mathfrak{sl}}_3$ is a direct sum of rank-one chains, each of which is a $V_n$ from the ${\mathfrak{sl}}_2$ classification. The hexagonal weight diagram of $V_{a,b}$ projects along any one simple-root direction to a union of ${\mathfrak{sl}}_2$-ladders, recovering the rank-one picture as a slice.

• Lie algebra representation 07.06.01. The general definition of a Lie algebra representation specialises to the rank-two case here: every finite-dimensional representation of ${\mathfrak{sl}}_3\mathbb{C}$ decomposes as a direct sum of irreducible $V_{a,b}$, parametrised by integral dominant highest weights. The same definitional framework that gave the $V_n$ classification for ${\mathfrak{sl}}_2$ (07.06.11) gives the $V_{a,b}$ classification here, with the same Casimir-element and weight-space machinery propagating to higher rank.

• Root system 07.06.03. The root system of ${\mathfrak{sl}}_3\mathbb{C}$ is $A_2$, with six roots arranged in a regular hexagon. The Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$, and the Weyl group is $S_3$. The general theory of root systems classifies the rank-two simply-laced case as exactly $A_2$, and the ${\mathfrak{sl}}_3$ representation theory of this unit is the canonical realisation. The fundamental weights ${\lambda }_1,{\lambda }_2$ are the dual basis to the simple coroots and parametrise the irreducible representations.

• Verma module 07.06.06. The Verma module $M(a,b)=U({\mathfrak{sl}}_3){\otimes }_{U(\mathfrak{b})}{\mathbb{C}}_{a{\lambda }_1+b{\lambda }_2}$ is the infinite-dimensional universal highest-weight module, and the finite-dimensional irreducible $V_{a,b}$ is the unique irreducible quotient. The kernel making the quotient finite-dimensional is generated by $F_{{\alpha }_1}^{a+1}{v}_{\lambda }$ and $F_{{\alpha }_2}^{b+1}{v}_{\lambda }$ (the Serre relations applied to the integrality conditions), exactly mirroring the ${\mathfrak{sl}}_2$ case where the kernel was generated by $F^{n+1}{v}_n$.

• Weyl character formula 07.06.07. The character of $V_{a,b}$ is computed by the Weyl character formula as a ratio of alternating sums over the Weyl group $S_3$: ${\chi }_{V_{a,b}}={\sum }_{w\in S_3}(-1{)}^{\ell (w)}{e}^{w(\lambda +\rho )}/{\sum }_{w\in S_3}(-1{)}^{\ell (w)}{e}^{w\rho }$, where $\ell$ is the length of $w$ and the denominator is the Weyl denominator. The rank-two character is no longer a single rational expression in a single variable but a ratio of three-term polynomials in two formal variables, reflecting the two Cartan directions.

• Weyl dimension formula 07.06.08. The general Weyl dimension formula specialises to $\dim V_{a,b}=(a+1)(b+1)(a+b+2)/2$ for ${\mathfrak{sl}}_3$, a polynomial of degree three in $(a,b)$ generalising the ${\mathfrak{sl}}_2$ formula $\dim V_n=n+1$. The product over positive roots picks up three factors, one per positive root of $A_2$, exactly the rank-two pattern.

• Borel-Weil theorem 07.06.09. The Borel-Weil theorem realises every $V_{a,b}$ as $H^0({\mathrm{Fl}}_3(\mathbb{C}),{\mathcal{L}}_{a,b})$, the space of holomorphic sections of a line bundle on the full flag variety of ${\mathbb{C}}^3$. The flag variety is a six-dimensional complex projective variety, and the line bundles are indexed by pairs of non-negative integers exactly matching the highest-weight classification. This is the rank-two generalisation of the ${\mathfrak{sl}}_2$ realisation on $\mathbb{C}{P}^1$.

• Schur-Weyl duality 07.05.04. The $V_{a,b}$ classification carried out here through the highest-weight machinery is recovered by Schur-Weyl duality applied to $V={\mathbb{C}}^3$: each Weyl module ${\mathbb{S}}^{\lambda }{\mathbb{C}}^3$ for a partition $\lambda$ with $\ell (\lambda )\le 3$ is the irreducible $\mathrm{GL}(3,\mathbb{C})$-representation of highest weight $\lambda$, which descends to $V_{a,b}$ with $a={\lambda }_1-{\lambda }_2$ and $b={\lambda }_2-{\lambda }_3$. Schur-Weyl is therefore the route by which the $S_n$-side combinatorics of Young symmetrisers and Specht modules produces the same rank-two classification this unit derives from root systems and Cartan integers, with the Schur polynomial $s_{\lambda }(x_1,x_2,x_3)$ on the symmetric-function side equal to the character of $V_{a,b}$ on the Lie-algebraic side. The agreement of the two derivations — partition-combinatorial via tensor powers, Cartan-integral via highest weights — is one of the cleanest concrete instances of the Schur-Weyl bridge.

• Casimir element 07.06.10. Direct anchor. The dedicated Casimir unit develops the general construction $C=X_a{X}^a\in U(\mathfrak{g})$, the centrality proof via the invariance of the Killing form, and the eigenvalue formula $c_{V_{\lambda }}=⟨\lambda ,\lambda +2\rho ⟩$ in arbitrary rank. The present unit is the canonical rank-two worked example: with $\lambda =a{\lambda }_1+b{\lambda }_2$ and $\rho ={\lambda }_1+{\lambda }_2$, the quadratic Casimir acts on $V_{a,b}$ by $⟨\lambda ,\lambda +2\rho ⟩$, which on the adjoint $V_{1,1}$ specialises to the scalar $6$. Unlike the rank-one case, the quadratic Casimir alone no longer separates all irreducibles: the joint spectrum of the quadratic and cubic Casimirs (the two generators of the Harish-Chandra centre $Z(U({\mathfrak{sl}}_3\mathbb{C}))$) gives the full infinitesimal character ${\chi }_{\lambda }$, with the cubic Casimir as the second generator predicted by the general Harish-Chandra theorem $Z(U(\mathfrak{g}))\cong S(\mathfrak{h}{)}^W$ in rank two. The Casimir-centrality proof we run in this unit is a direct instance of the general invariance-of-Killing-form argument given in 07.06.10.

Historical & philosophical context [Master]

The representation theory of ${\mathfrak{sl}}_3\mathbb{C}$ entered the mathematical record as the rank-two case of Wilhelm Killing's 1888-90 Mathematische Annalen series Die Zusammensetzung der stetigen endlichen Transformations-Gruppen (vols 31, 33, 34, 36) ^[pending], where Killing identified the type-$A$ series and labelled the rank-two simply-laced case as $A_2$. Killing's classification was incomplete and had errors that Élie Cartan corrected in his 1894 doctoral thesis Sur la structure des groupes de transformations finis et continus (Faculté des Sciences de Paris) ^[pending], where the modern $A_n$ family of simple Lie algebras was given with ${\mathfrak{sl}}_3=A_2$ as the rank-two base case. Cartan's thesis already contained the essential structural data: the Cartan subalgebra of dimension two, the root system $A_2$ with six roots arranged hexagonally, the Cartan matrix, and the simple-root labelling. The thesis did not contain the full theory of finite-dimensional representations — that came later.

The representation theory itself was developed by Hermann Weyl in the 1925-26 papers Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen (Math. Z. 23, 271-309 and Math. Z. 24, 328-376, 377-395, 789-791) ^[pending], establishing the character formula and the dimension formula in full generality. For ${\mathfrak{sl}}_3$ specifically, the dimension formula $\dim V_{a,b}=(a+1)(b+1)(a+b+2)/2$ is the rank-two specialisation, and Weyl computed it explicitly as an example in his original papers. The unitarian-trick proof of complete reducibility applies in this case via the compact real form $\mathfrak{su}(3)\subseteq {\mathfrak{sl}}_3\mathbb{C}$, whose Lie group $\mathrm{SU}(3)$ is connected, simply connected, and compact, giving an averaging argument over Haar measure to produce invariant Hermitian inner products. The Cartan-Eilenberg 1948 Annals of Math. paper Cohomology theory of Lie groups and Lie algebras ^[pending] situated this representation theory inside a broader cohomological framework.

The multiplicity-counting machinery for weight spaces — including the multiplicity-two zero weight of the adjoint representation — was put on a closed-form recursive footing by Hans Freudenthal in Zur Berechnung der Charaktere der halbeinfachen Lieschen Gruppen (Indagationes Mathematicae 16, 369-376, 1954) ^[pending]. The Freudenthal multiplicity formula gives a recursive identity computing $m_{\mu }$ from the multiplicities of weights strictly above $\mu$ in the partial order, and the rank-two specialisation gives an effective algorithm for the multiplicities in $V_{a,b}$. An alternative derivation, organising the character as an inclusion-exclusion over the Weyl group, was given by Robert Steinberg in A general Clebsch-Gordan theorem (Trans. Amer. Math. Soc. 100, 392-400, 1961) ^[pending], which yields the same hexagonal weight structure via a different combinatorial route.

The physics-mathematics dictionary places ${\mathfrak{sl}}_3\mathbb{C}$ at the centre of twentieth-century particle physics. Murray Gell-Mann's and Yuval Ne'eman's 1961 eightfold way identified the eight observed light mesons and eight light baryons with the adjoint representation $V_{1,1}$ of $\mathfrak{su}(3)={\mathfrak{sl}}_3\mathbb{C}$ acting on $V\otimes V^{\ast }=V_{1,1}\oplus \mathbb{C}$, and the three quark flavours (up, down, strange) with the fundamental $V={\mathbb{C}}^3$. The Cabibbo-Kobayashi-Maskawa mixing matrix and the eight gluons of quantum chromodynamics are direct manifestations of $\mathfrak{su}(3)$ representation theory in the Standard Model of particle physics. The hexagonal weight diagram of $V_{1,1}$ became iconic as the diagrammatic record of the eight pseudoscalar mesons, with the multiplicity-two zero weight corresponding to the $\eta$ and ${\pi }^0$ mesons sharing the same quantum numbers. The mathematical and physical pictures are the same picture.

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