07.06.25 · representation-theory / lie-algebraic

Weyl construction of the classical-group irreducibles

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Anchor (Master): Hermann Weyl 1939 *The Classical Groups: Their Invariants and Representations* (Princeton) Ch. III-VI; Richard Brauer 1937 *On algebras which are connected with the semisimple continuous groups* (Ann. of Math. 38, 857-872) — the Brauer-algebra centraliser of $\mathrm{O}(V)$ and $\mathrm{Sp}(V)$ on $V^{\otimes d}$; Cartan 1913 *Les groupes projectifs qui ne laissent invariante aucune multiplicité plane* (Bull. SMF 41) — the spin representations; Fulton-Harris *Representation Theory* §§16-19; Goodman-Wallach Ch. 10; Howe 1989 *Remarks on classical invariant theory* (Trans. AMS 313) — the dual-pair organising principle

Intuition Beginner

For the matrices that preserve only volume — the algebra $sl_{n}$ — building every irreducible representation is a clean story. You take the standard space $V = C^{n}$ , form a tensor power, and cut it into pieces by symmetry type, one piece for each Young diagram. Each piece is already a single irreducible building block. The unit on Schur-Weyl duality tells exactly how this cutting works.

The classical groups beyond $sl_{n}$ preserve more than volume. The symplectic algebra $sp_{2 n}$ preserves a skew pairing of vectors, and the orthogonal algebra $so_{m}$ preserves a length, a quadratic form. That extra preserved structure is a gift and a complication at once. It is a gift because it lets you take two vectors and pair them into a single number, an operation that respects the group action. It is a complication because the pieces you got from pure symmetry are no longer the smallest building blocks: each one still contains smaller pieces hiding inside.

The Weyl construction is the recipe that finishes the job. After cutting a tensor power by symmetry type, you throw away everything that the invariant pairing can detect — every tensor that has a nonzero "trace" with respect to the form. What survives is the traceless, or primitive, part, and that part is a genuine irreducible building block. For the symplectic case this always works. For the orthogonal case it works too, with one famous exception: the spin representations, which live a half-step outside the world of tensors and need the language of Clifford algebras to reach.

Visual Beginner

A picture in three stacked panels. The top panel shows the type- $A$ story: a big box labelled "tensor power of $V$ " splits cleanly into colored blocks, one per Young diagram, each block a finished irreducible — no leftovers.

The middle panel shows the symplectic and orthogonal story: the same big box splits into the same colored blocks, but now each block has a smaller shaded core inside it (the traceless part) wrapped in a thin shell (the part detected by the form). An arrow labelled "contraction" points from each block out to a smaller box (the tensor power with two slots removed), showing where the shell goes. The bottom panel zooms into one orthogonal exterior-power box $Λ^{k} V$ and shows it breaking into two pieces side by side: a harmonic core $Λ_{0}^{k} V$ and a copy of the smaller $Λ^{k - 2} V$ .

The single idea carried by all three panels: the invariant bilinear form gives you a way to shrink a tensor by two slots, and the irreducible pieces are exactly the tensors that cannot be shrunk this way. Symmetry type sets the outer shape; the trace-free condition carves out the irreducible inside it.

Worked example Beginner

Take the standard representation of $sp_{4}$ , the space $V = C^{4}$ with a skew pairing $ω$ . Form the second tensor power of $V$ with itself, which has dimension $16$ . Split it by symmetry into the symmetric part $Sym^{2} V$ of dimension $10$ and the exterior part $Λ^{2} V$ of dimension $6$ .

The symmetric part $Sym^{2} V$ stays in one piece: it is already a single irreducible of $sp_{4}$ , of dimension $10$ . This is the adjoint representation — the algebra $sp_{4}$ acting on itself, since a symplectic algebra is the same size as the symmetric square of its standard space.

The exterior part $Λ^{2} V$ is where the form bites. The skew pairing $ω$ is itself an element of $Λ^{2} V$ , and it is fixed by the whole group. So $Λ^{2} V$ contains a one-dimensional fixed line spanned by $ω$ . Pull that line out. What remains is a $6 - 1 = 5$ -dimensional piece, the traceless part of $Λ^{2} V$ , and that piece is a single irreducible of $sp_{4}$ , of dimension $5$ .

So the symplectic answer is $Λ^{2} V = Λ_{0}^{2} V \oplus C ω$ , splitting $6 = 5 + 1$ , while the type- $A$ answer would have left $Λ^{2} V$ whole. The number $5$ is the dimension that matters: it is the standard representation of $so_{5}$ . The same five-dimensional irreducible showing up on both sides is the first visible sign of the isomorphism $so_{5} ≅ sp_{4}$ .

Check your understanding Beginner

Exercise (easy, multiple choice).

In the Weyl construction for $sp_{2 n}$ and $so_{m}$ , what extra step is needed compared with the $sl_{n}$ case?

A. Nothing extra; the symmetry pieces are already the answer
B. Remove the part detected by the invariant bilinear form (the traceless part survives)
C. Add a Clifford algebra to every piece
D. Replace tensor powers by symmetric powers only

Hint

The defining feature of $sp$ and $so$ is that they preserve a bilinear form, which the volume-only algebra $sl_{n}$ does not.

Answer

B. The invariant form gives contraction maps that shrink a tensor by two slots; the irreducible piece is the traceless (primitive) part that all contractions kill. Feedback-correct: cutting by symmetry type is only the first step, and the trace-free condition carves the irreducible out of each symmetry block. Feedback-wrong: A is the $sl_{n}$ answer, where blocks are already irreducible; C confuses the tensor construction with the separate spin construction; D drops the antisymmetric pieces that carry the fundamental representations.

Exercise (medium, true-false).

True or false: the Weyl tensor construction produces every finite-dimensional irreducible representation of $so_{m}$ .

Hint

Recall the one famous exception mentioned in the Intuition section. Think about whether a half-integer can be a partition entry.

Answer

False. The tensor construction produces exactly the irreducibles whose highest weight is a partition (integer entries). The spin representations of $so_{m}$ have half-integer highest weights and live outside the tensor powers; they require the Clifford algebra and the spin cover. Feedback-correct: the symplectic case has no such gap, but the orthogonal case misses the spin (and half-spin) representations. Feedback-wrong: answering "true" forgets the spin representations, which is the central subtlety of the orthogonal case.

Formal definition Intermediate+

Fix a finite-dimensional complex vector space $V$ carrying a non-degenerate bilinear form $B$ . Two cases organise the whole theory: $B$ skew-symmetric, with $V = C^{2 n}$ and the invariance group $Sp (V) = Sp_{2 n} C$ and Lie algebra $sp_{2 n} C$ ; or $B$ symmetric, with $V = C^{m}$ and the invariance group $O (V) = O_{m} C$ and Lie algebra $so_{m} C$ . In each case the group preserves $B$ , so it preserves the contraction maps built from $B$ .

The form $B$ defines, for each ordered pair of tensor slots $(p, q)$ with $1 \leq p < q \leq d$ , a contraction $C_{pq} : V^{\otimes d} \to V^{\otimes (d - 2)}$ that pairs the $p$ -th and $q$ -th factors using $B$ and deletes them: $$ C_{pq}(v_1 \otimes \cdots \otimes v_d) = B(v_p, v_q), v_1 \otimes \cdots \widehat{v_p} \cdots \widehat{v_q} \cdots \otimes v_d. $$ Because $B$ is invariant, each $C_{pq}$ is a homomorphism of representations of $Sp (V)$ or $O (V)$ . The subspace of traceless (or primitive, or in the orthogonal case harmonic) tensors is the common kernel $$ V^{\langle d \rangle} = \bigcap_{1 \le p < q \le d} \ker C_{pq} \subseteq V^{\otimes d}. $$

The symmetric group $S_{d}$ acts on $V^{\otimes d}$ by permuting factors, and this action commutes with the classical-group action and preserves $V^{⟨ d ⟩}$ . Given a partition $λ$ of $d$ , let $c_{λ} \in C [S_{d}]$ be a Young symmetriser and $S_{λ} (V) = c_{λ} \cdot V^{\otimes d}$ the associated $GL (V)$ Weyl module (the Schur functor of 07.05.04). The symplectic/orthogonal Schur functor is the image of the traceless tensors under the Young symmetriser, equivalently the traceless part of the Weyl module: $$ \mathbb{S}{[\lambda]}(V) = \mathbb{S}\lambda(V) \cap V^{\langle d \rangle} = c_\lambda \cdot V^{\langle d \rangle}. $$

A partition $λ = (λ_{1} \geq λ_{2} \geq \dots)$ is admissible for $sp_{2 n}$ when its number of rows satisfies $ℓ (λ) \leq n$ , and for $so_{m}$ when the sum of the first two column lengths $λ_{1}^{'} + λ_{2}^{'} \leq m$ (so that the diagram fits the orthogonal constraint). The highest weight read off a partition $λ$ is $λ_{1} L_{1} + λ_{2} L_{2} + \dots$ in the standard weight basis $L_{i}$ of the Cartan subalgebra, exactly as for $gl$ and $sl$ , restricted to the classical Cartan.

Counterexamples to common slips

The Weyl module $S_{λ} (V)$ for $GL (V)$ is irreducible as a $GL (V)$ -module, but not as a $Sp (V)$ - or $O (V)$ -module: restricting to the smaller group, it picks up extra invariants visible to $B$ , so it splits. The intersection with $V^{⟨ d ⟩}$ is needed to land on a single irreducible.
The traceless condition is not the same as taking the antisymmetric or symmetric part. For $sp_{4}$ the symmetric square $Sym^{2} V$ is already traceless (the skew form pairs antisymmetrically, so it contracts the antisymmetric slot to zero), while the exterior square $Λ^{2} V$ is not.
For $so_{m}$ the construction misses the spin representations entirely. A highest weight with a half-integer entry, such as $\frac{1}{2} (L_{1} + \dots + L_{n})$ , is never a partition and so never appears in any $V^{\otimes d}$ ; it requires the Clifford-algebra spin module.

Key theorem with proof Intermediate+

Theorem (Weyl construction of classical-group irreducibles). Let $V$ carry a non-degenerate symplectic form (case $sp_{2 n}$ , $V = C^{2 n}$ ) or a non-degenerate symmetric form (case $so_{m}$ , $V = C^{m}$ ). For every admissible partition $λ$ the space $S_{[λ]} (V) = S_{λ} (V) \cap V^{⟨ d ⟩}$ is either zero or an irreducible representation of the classical Lie algebra with highest weight $λ_{1} L_{1} + λ_{2} L_{2} + \dots$ . In the symplectic case $S_{[λ]} (V) \neq = 0$ exactly when $ℓ (λ) \leq n$ , and these exhaust the finite-dimensional irreducibles of $sp_{2 n} C$ . In the orthogonal case the non-zero $S_{[λ]} (V)$ realise every irreducible of $so_{m} C$ whose highest weight is a partition; the spin and half-spin representations, with half-integer highest weights, are not obtained.

Proof. The argument has three movements: the contractions are equivariant, so $V^{⟨ d ⟩}$ is a subrepresentation; a highest-weight vector lives inside it; and a centraliser (double-commutant) count forces irreducibility.

Step 1: the traceless tensors form a subrepresentation, and a highest-weight vector survives. Each contraction $C_{pq}$ is built from the invariant form $B$ , hence commutes with the action of the classical group, so $V^{⟨ d ⟩}$ is a subrepresentation of $V^{\otimes d}$ . The $S_{d}$ -action commutes with the group action, so $S_{λ} (V) \cap V^{⟨ d ⟩}$ is again a subrepresentation. To see it is non-zero in the admissible range, take the standard highest-weight tensor of $S_{λ} (V)$ , the image under $c_{λ}$ of $e_{1}^{\otimes λ_{1}} \otimes e_{2}^{\otimes λ_{2}} \otimes \dots$ where $e_{1}, e_{2}, \dots$ are the first basis vectors spanning a maximal isotropic flag. Because these basis vectors are mutually isotropic and orthogonal in the chosen polarisation of $B$ (each $B (e_{i}, e_{j}) = 0$ for the indices appearing), every contraction $C_{pq}$ annihilates this tensor. So the highest-weight vector lies in $V^{⟨ d ⟩}$ , and the weight it carries is $λ_{1} L_{1} + λ_{2} L_{2} + \dots$ .

Step 2: admissibility. The vanishing in the inadmissible range is a Cartan-rank constraint. For $sp_{2 n}$ the Cartan subalgebra has rank $n$ , so a partition with more than $n$ rows has a highest weight outside the dominant cone after the symplectic identifications $L_{i} \leftrightarrow - L_{i + n}$ , and the corresponding symmetriser composed with the traceless projection gives zero. For $so_{m}$ the orthogonal relations $Λ^{k} V ≅ Λ^{m - k} V$ (Hodge duality) fold the diagram, and partitions whose first two columns overstep $m$ vanish.

Step 3: irreducibility by the double commutant. On $V^{\otimes d}$ the algebra generated by the classical group and its centraliser are full mutual commutants — this is the orthogonal/symplectic Schur-Weyl duality, where the centraliser is the Brauer algebra $B_{d} (δ)$ with parameter $δ = m$ (orthogonal) or $δ = - 2 n$ (symplectic), replacing the group algebra $C [S_{d}]$ of the type- $A$ story in 07.05.04. The double-commutant theorem then matches irreducible classical-group summands of $V^{⟨ d ⟩}$ with irreducible modules of the Brauer algebra, and the Young symmetriser $c_{λ}$ projects onto a single such isotypic piece. Hence $S_{[λ]} (V)$ , when non-zero, is irreducible. Its highest weight is the one computed in Step 1, and varying $λ$ over admissible partitions exhausts the partition-weight irreducibles. The half-integer-weight spin modules are absent because no power $V^{\otimes d}$ contains a vector of half-integer weight. $□$

Bridge. This theorem builds toward the full highest-weight classification for all semisimple Lie algebras and appears again in the spin-geometry and invariant-theory chapters, where the missing spin representations are constructed by hand. The foundational reason the construction needs the extra contraction step is exactly the presence of the invariant form: where $sl_{n}$ preserves only the volume and the Weyl module $S_{λ} (V)$ is already irreducible (the central insight of 07.05.04), the symplectic and orthogonal algebras preserve a pairing whose contractions detect proper subrepresentations, and the trace-free condition is the bridge that removes them. This is exactly the rank-one-per-direction integrality of 07.06.11 and 07.06.12 re-expressed in tensor language: the highest weight read off a Young diagram is dominant integral, and the worked $sl_{2}$ and $sl_{3}$ ladders generalise to the classical Cartan by the same weight-counting. Putting these together, the Brauer algebra is dual to the classical group on $V^{\otimes d}$ in the precise sense that the symmetric-group duality of Schur-Weyl generalises to a larger diagram algebra; the contraction maps are the diagrammatic arcs that the Brauer algebra adds to the permutation diagrams of $S_{d}$ .

Exercises Intermediate+

Exercise 1 (easy, numeric).

For $so_{5}$ with $V = C^{5}$ , compute the dimension of the harmonic part $Λ_{0}^{2} V$ of the exterior square, given that $Λ^{2} V$ has dimension $(2 5)$ and the orthogonal splitting removes a copy of $Λ^{0} V$ .

Hint

The orthogonal splitting is $Λ^{2} V = Λ_{0}^{2} V \oplus Λ^{0} V$ only when a scalar contraction applies; here $Λ^{2}$ has no scalar to remove, so $Λ^{2} V$ is already harmonic. Compute $(2 5)$ .

Answer

$10$ . The exterior square $Λ^{2} V$ has dimension $(2 5) = 10$ , and for $Λ^{2}$ the orthogonal contraction $C_{12}$ maps into $Λ^{0} V$ but the antisymmetric form makes the symmetric quadratic contraction vanish, so $Λ^{2} V = Λ_{0}^{2} V$ is already harmonic of dimension $10$ . This is the adjoint representation of $so_{5}$ , matching $dim so_{5} = (2 5) = 10 = dim sp_{4}$ . Rubric: full credit for $10$ with the identification as the adjoint.

Exercise 3 (medium, symbolic).

Decompose $V \otimes V$ for $sp_{2 n}$ ( $V = C^{2 n}$ ) into irreducibles, identifying each piece by its symmetry type and its traceless content.

Hint

Split into $Sym^{2} V$ and $Λ^{2} V$ . The symplectic form lives in $Λ^{2} V$ . Check which piece is already traceless.

Answer

$V \otimes V = Sym^{2} V \oplus Λ_{0}^{2} V \oplus C$ , of dimensions $(2 2 n + 1) + ((2 2 n) - 1) + 1 = (2 n^{2} + n) + (2 n^{2} - n - 1) + 1 = 4 n^{2}$ . The symmetric square $Sym^{2} V ≅ sp_{2 n}$ is the adjoint, irreducible of highest weight $2 L_{1}$ and already traceless (the skew form contracts it to zero). The exterior square $Λ^{2} V$ contains the invariant line $C ω$ spanned by the symplectic form; removing it leaves the traceless $Λ_{0}^{2} V$ , irreducible of highest weight $L_{1} + L_{2}$ . Rubric: full credit for the three pieces with the symmetric square identified as the adjoint and the invariant line pulled from the exterior square.

Exercise 4 (medium, short-answer).

Explain why the symmetric square $Sym^{2} V$ is already irreducible for $sp_{2 n}$ but splits for $so_{m}$ , while the exterior square $Λ^{2} V$ does the reverse.

Hint

A contraction $C_{12}$ uses the invariant form. A skew form pairs antisymmetric slots; a symmetric form pairs symmetric slots. Which square does each contraction act on with a non-zero image?

Answer

The contraction $C_{12}$ inserts the form $B$ into the two factors. When $B$ is skew (symplectic), $C_{12}$ is non-zero only on the antisymmetric part, so it detects an invariant in $Λ^{2} V$ (namely $ω$ ) and kills $Λ^{2} V$ 's scalar piece, while $Sym^{2} V$ is automatically traceless and stays whole as the adjoint. When $B$ is symmetric (orthogonal), $C_{12}$ is non-zero only on the symmetric part, so it detects the invariant quadratic form inside $Sym^{2} V$ (the line $C B$ ), splitting off a scalar and leaving the harmonic $Sym_{0}^{2} V$ , while $Λ^{2} V ≅ so_{m}$ stays whole as the adjoint. Rubric: full credit for tying the parity of the form to which square the contraction acts on.

Exercise 5 (medium, symbolic).

Show the splitting $Λ^{k} V = Λ_{0}^{k} V \oplus Λ^{k - 2} V$ for $so_{m}$ at the level of dimensions for $m = 5$ , $k = 2$ , and explain the geometric content.

Hint

The contraction with the quadratic form sends $Λ^{k} V \to Λ^{k - 2} V$ , and wedging with the form is a section going back. For $k = 2$ , $m = 5$ check whether the contraction is non-zero.

Answer

For $so_{5}$ , $k = 2$ : $Λ^{2} V$ has dimension $(2 5) = 10$ , and $Λ^{0} V$ has dimension $1$ . The orthogonal contraction $C : Λ^{2} V \to Λ^{0} V$ is built from the symmetric form acting on an antisymmetric tensor, which vanishes identically, so the splitting is $Λ^{2} V = Λ_{0}^{2} V \oplus 0$ , i.e. $Λ^{2} V$ is already harmonic of dimension $10$ . The instance with a non-zero contraction is $Sym^{2} V = Sym_{0}^{2} V \oplus C$ , $15 = 14 + 1$ , where the quadratic form is the scalar removed. Geometrically, contraction with the metric and wedging with the metric form an $sl_{2}$ -action (a Howe dual pair) on the full exterior or symmetric algebra, and the harmonic pieces are the lowest-weight vectors of this $sl_{2}$ . Rubric: full credit for the dimension bookkeeping and the recognition that the antisymmetric contraction vanishes here, plus the $sl_{2}$ /Howe-pair remark.

Exercise 6 (hard, short-answer).

State precisely why the spin representations of $so_{m}$ cannot arise as any $S_{[λ]} (V)$ , and name the construction that does produce them.

Hint

Every weight of $V^{\otimes d}$ is an integer combination of the $L_{i}$ . What is the highest weight of a spin representation?

Answer

Every weight occurring in $V^{\otimes d}$ is a sum of $d$ weights each of the form $\pm L_{i}$ , hence an integer combination of the $L_{i}$ lying in the root lattice's integer span. The spin representation of $so_{2 n + 1}$ has highest weight $\frac{1}{2} (L_{1} + L_{2} + \dots + L_{n})$ , and the two half-spin representations of $so_{2 n}$ have highest weights $\frac{1}{2} (L_{1} + \dots + L_{n - 1} \pm L_{n})$ — all with half-integer entries, lying in the weight lattice but not the root lattice. No tensor power contains such a weight, so no $S_{[λ]} (V)$ realises them. They are constructed as modules over the Clifford algebra $Cl (V, B)$ : the spinor space is an irreducible Clifford module, $so_{m}$ embeds in $Cl (V)$ via the quadratic elements $e_{i} e_{j}$ , and for $m = 2 n$ the volume element splits the spinor space into the two half-spin pieces. Rubric: full credit for the half-integer-weight obstruction plus naming the Clifford-algebra spinor construction.

Exercise 7 (hard, short-answer).

Verify the exceptional isomorphism $so_{5} ≅ sp_{4}$ at the level of standard and adjoint representations.

Hint

Match $dim so_{5}$ with $dim sp_{4}$ , and identify the five-dimensional $sp_{4}$ -irreducible $Λ_{0}^{2} (C^{4})$ with the standard $so_{5}$ -representation $C^{5}$ .

Answer

Both algebras have dimension $10$ : $dim so_{5} = (2 5) = 10$ and $dim sp_{4} = 2 \cdot 2^{2} + 2 = 10$ . The Dynkin diagrams agree: $B_{2} = C_{2}$ , two nodes joined by a double bond, the only rank-two non-simply-laced diagram up to the arrow's direction (which $B_{2}$ and $C_{2}$ exchange). On the standard side, $sp_{4}$ acts on $Λ_{0}^{2} (C^{4})$ , the five-dimensional traceless exterior square, and this carries a non-degenerate symmetric form (the wedge pairing $Λ^{2} \otimes Λ^{2} \to Λ^{4} ≅ C$ restricted to the traceless part), so $sp_{4}$ acts through $so_{5}$ on $C^{5} = Λ_{0}^{2} (C^{4})$ . The adjoint representations match: $so_{5} = Λ^{2} (C^{5})$ of dimension $10$ and $sp_{4} = Sym^{2} (C^{4})$ of dimension $10$ are isomorphic as the common adjoint. The spin representation of $so_{5}$ , of dimension $4$ , corresponds under the isomorphism to the standard representation $C^{4}$ of $sp_{4}$ — a tensor irreducible on the symplectic side but a spin irreducible on the orthogonal side, which is why $C^{4}$ does not appear among the orthogonal tensor constructions. Rubric: full credit for the dimension match, the $B_{2} = C_{2}$ diagram identity, and the standard/spin correspondence.

Advanced results Master

Theorem (harmonic decomposition for $O (V)$ ). For the orthogonal group, the tensor power decomposes as a direct sum over traceless pieces tagged by the number of contracted pairs: $$ V^{\otimes d} = \bigoplus_{f \ge 0} ; \bigoplus_{|\mu| = d - 2f} ; \big( V^{\langle d-2f \rangle}{\mu} \big)^{\oplus N{\mu, f}}, $$ where the inner sum runs over partitions $μ$ and the multiplicities $N_{μ, f}$ are computed from the Brauer-algebra branching. Equivalently, the algebra of $O (V)$ -invariants in $⨁_{d} V^{\otimes d}$ is generated by the form $B$ and its contractions — the first fundamental theorem of invariant theory for the orthogonal group.

This is the orthogonal analogue of the Schur-Weyl decomposition of 07.05.04. Where the symmetric group $S_{d}$ indexed the type- $A$ multiplicities by partitions of $d$ , the Brauer algebra indexes the orthogonal and symplectic multiplicities by partitions of $d - 2 f$ for every number $f$ of contracted pairs. Each application of a contraction $C_{pq}$ lowers the tensor degree by $2$ , and the harmonic pieces are the kernels at each level. The decomposition is governed by an $sl_{2}$ -triple of "raise degree by wedging with $B$ " and "lower degree by contracting with $B$ ", the Howe dual pair $(O (V), sl_{2})$ .

Theorem (symplectic primitive decomposition). For the symplectic group, contraction and the wedge with the form $ω$ generate an action of $sl_{2}$ (the Lefschetz-style triple) on $Λ^{∙} V$ , and the exterior algebra decomposes into primitive pieces $$ \Lambda^k V = \bigoplus_{j \ge 0} \omega^j \wedge \Lambda^{k - 2j}_0 V, $$ where $Λ_{0}^{k} V = ker (C : Λ^{k} V \to Λ^{k - 2} V)$ is the primitive part, irreducible of highest weight $L_{1} + \dots + L_{k}$ for $k \leq n$ . The $Λ_{0}^{k} V$ for $1 \leq k \leq n$ are precisely the fundamental representations of $sp_{2 n}$ .

This is the hard-Lefschetz pattern made representation-theoretic: the symplectic form $ω$ acts as the Lefschetz operator $L$ , the contraction acts as $Λ$ , and their commutator is the degree-counting $H$ . The primitive cohomology of Kähler geometry and the symplectic fundamental representations are the same $sl_{2}$ -decomposition seen in two settings. The $n$ fundamental representations of $sp_{2 n}$ are the primitive exterior powers $Λ_{0}^{k} V$ , exactly $n$ of them, matching the rank, just as the $n$ fundamentals of $sl_{n + 1}$ were the full exterior powers in 07.06.12.

Theorem (orthogonal exterior splitting and self-duality). For $so_{m}$ the exterior powers split as $Λ^{k} V = Λ_{0}^{k} V \oplus Λ^{k - 2} V$ with $Λ_{0}^{k} V$ irreducible for $k < m /2$ , and Hodge duality gives $Λ_{0}^{k} V ≅ Λ_{0}^{m - k} V$ . When $m = 2 n$ is even, the middle exterior power $Λ_{0}^{n} V = Λ^{n} V$ splits further into self-dual and anti-self-dual halves $Λ_{+}^{n} V \oplus Λ_{-}^{n} V$ under the Hodge star, each irreducible of highest weight $L_{1} + \dots + L_{n - 1} \pm L_{n}$ .

The middle-dimensional splitting is the source of the two half-spin-adjacent representations on the tensor side, and it is the combinatorial shadow of the fact that the Dynkin diagram $D_{n}$ forks at the right end into two nodes. The self-dual and anti-self-dual forms of four-dimensional geometry, the decomposition of the curvature tensor, and the two half-spin chiralities all descend from this single splitting of the middle exterior power.

Theorem (Howe duality / first fundamental theorem). The commutant of $Sp (2 n)$ acting diagonally on $Sym^{∙} (C^{2 n} \otimes C^{k})$ is generated by an $so_{2 k}$ -action, and dually the commutant of $O (m)$ on $Sym^{∙} (C^{m} \otimes C^{k})$ is an $sp_{2 k}$ -action. These reductive dual pairs $(Sp (2 n), O (2 k))$ and $(O (m), Sp (2 k))$ organise all the contractions and harmonic projections of the Weyl construction into a single joint-multiplicity-free decomposition. ^{[Howe 1989]}

Howe duality is the structural summit of the whole construction: the seemingly ad-hoc contraction maps are the infinitesimal action of the dual partner in a reductive pair, and the traceless tensors are the joint highest-weight vectors. The Weyl construction, the harmonic decomposition, and the theta correspondence of automorphic forms are three faces of the same dual-pair phenomenon.

Synthesis. The Weyl construction is the foundational reason that the entire partition-indexed family of classical-group irreducibles lives inside tensor powers of the standard representation, and it is exactly the type- $A$ Schur-functor story of 07.05.04 corrected by one structural feature: the invariant bilinear form. This is the central insight that unifies the symplectic and orthogonal cases — both add contraction maps to the symmetric-group duality, and the trace-free tensors are the irreducibles — and it generalises the rank-one and rank-two ladder constructions of 07.06.11 and 07.06.12 to the full classical Cartan by reading highest weights off Young diagrams. Putting these together, the Brauer algebra is dual to the classical group in the precise double-commutant sense that the group algebra of $S_{d}$ was dual to $GL (V)$ , and the contraction arcs are the extra diagrammatic generators; this duality is exactly the Howe dual-pair structure restricted to a single tensor degree.

The bridge from the tensor world to the rest of representation theory is the deliberate gap the construction leaves: the spin and half-spin representations, with their half-integer highest weights, sit outside every tensor power and demand the Clifford algebra, so the Weyl construction both realises every partition-weight irreducible and points precisely at what it cannot reach. The hard-Lefschetz $sl_{2}$ that decomposes the exterior algebra into primitives is dual to the symplectic form, and the self-dual/anti-self-dual splitting of the middle exterior power for $so_{2 n}$ is the foundational reason the $D_{n}$ diagram forks — putting these together, the abstract Dynkin combinatorics and the concrete tensor decompositions are one structure seen from two sides.

Full proof set Master

Theorem (Weyl construction), proof. Given in full in the Intermediate-tier section: the contraction maps are equivariant, so the traceless tensors form a subrepresentation; the standard Young-symmetrised highest-weight tensor of $S_{λ} (V)$ is automatically traceless because the polarising basis is isotropic; and the orthogonal/symplectic Schur-Weyl duality with Brauer-algebra centraliser forces each $S_{[λ]} (V)$ to be irreducible by the double-commutant theorem. $□$

Proposition (the symplectic form spans the invariants of $Λ^{2} V$ ). For $V = C^{2 n}$ with symplectic form $ω$ , the space of $Sp (V)$ -invariants in $Λ^{2} V$ is the one-dimensional line $C ω$ , and the traceless complement $Λ_{0}^{2} V$ has dimension $(2 2 n) - 1 = 2 n^{2} - n - 1$ .

Proof. An invariant in $Λ^{2} V ≅ Λ^{2} V^{*}^{*}$ is an $Sp (V)$ -fixed skew bilinear form on $V^{*}$ , equivalently a fixed skew form on $V$ up to the identification $V ≅ V^{*}$ given by $ω$ itself. By the first fundamental theorem of invariant theory for the symplectic group, every $Sp (V)$ -invariant bilinear form on $V$ is a scalar multiple of $ω$ . So the invariant subspace of $Λ^{2} V$ is exactly $C ω$ , one-dimensional. The contraction $C : Λ^{2} V \to Λ^{0} V = C$ given by $v \land w \mapsto ω (v, w)$ is surjective with kernel $Λ_{0}^{2} V$ of dimension $(2 2 n) - 1$ , and the form $ω$ maps to $C (ω) = \sum_{i} ω (e_{i}, f_{i}) = n \neq = 0$ in a symplectic basis ${e_{i}, f_{i}}$ , confirming $Λ^{2} V = Λ_{0}^{2} V \oplus C ω$ . $□$

Proposition (the symmetric square is the symplectic adjoint). For $V = C^{2 n}$ symplectic, $Sym^{2} V ≅ sp_{2 n} C$ as representations of $sp_{2 n}$ , both of dimension $2 n^{2} + n$ , irreducible of highest weight $2 L_{1}$ .

Proof. The symplectic form $ω$ gives an isomorphism $V ≅ V^{*}$ , $v \mapsto ω (v, -)$ , hence $End (V) ≅ V \otimes V^{*} ≅ V \otimes V$ . Under this identification the Lie algebra $sp_{2 n} = {X : ω (X v, w) + ω (v, X w) = 0}$ corresponds to the symmetric tensors $Sym^{2} V$ : the condition that $X$ infinitesimally preserves the skew form $ω$ translates, after lowering an index by $ω$ , into the symmetry of the associated bilinear form. Counting dimensions: $dim Sym^{2} V = (2 2 n + 1) = 2 n^{2} + n = dim sp_{2 n}$ . The highest weight of $Sym^{2} V$ is $2 L_{1}$ (the square of the highest weight $L_{1}$ of $V$ ), which is the highest root of $C_{n}$ , identifying the module as the adjoint. It is traceless under the symplectic contraction because the contraction $C : Sym^{2} V \to C$ uses the skew form on a symmetric tensor and vanishes, so $Sym^{2} V = Sym_{0}^{2} V$ is already primitive and irreducible. $□$

Proposition (orthogonal adjoint and the exterior square). For $V = C^{m}$ orthogonal, $Λ^{2} V ≅ so_{m} C$ , of dimension $(2 m)$ , irreducible of highest weight $L_{1} + L_{2}$ (the highest root of $B_{n}$ or $D_{n}$ ), and already harmonic.

Proof. The symmetric form $B$ gives $V ≅ V^{*}$ , hence $End (V) ≅ V \otimes V$ . The Lie algebra $so_{m} = {X : B (X v, w) + B (v, X w) = 0}$ corresponds, after lowering an index by $B$ , to the antisymmetric tensors $Λ^{2} V$ : the condition that $X$ preserves the symmetric form translates into skew-symmetry of the associated bilinear form. Dimension count: $dim Λ^{2} V = (2 m) = dim so_{m}$ . The orthogonal contraction $C : Λ^{2} V \to C$ uses the symmetric form on an antisymmetric tensor and vanishes, so $Λ^{2} V$ is harmonic. Its highest weight is $L_{1} + L_{2}$ , the highest root, identifying it as the adjoint. $□$

Proposition ( $so_{5} ≅ sp_{4}$ via $Λ_{0}^{2}$ ). The map sending $sp_{4}$ to its action on $Λ_{0}^{2} (C^{4})$ realises an isomorphism $sp_{4} ≅ so_{5}$ , under which the standard $so_{5}$ -representation $C^{5}$ is $Λ_{0}^{2} (C^{4})$ and the spin representation of $so_{5}$ is the standard $C^{4}$ of $sp_{4}$ .

Proof. Let $W = Λ_{0}^{2} (C^{4})$ , of dimension $5$ by the symplectic-invariants proposition above. The wedge pairing $Λ^{2} V \otimes Λ^{2} V \to Λ^{4} V ≅ C$ is symmetric and non-degenerate; restricted to $W$ it remains non-degenerate (the radical would be an $sp_{4}$ -subrepresentation, and $W$ is irreducible). So $W$ carries an $sp_{4}$ -invariant non-degenerate symmetric form, giving a homomorphism $sp_{4} \to so (W) = so_{5}$ . Both algebras have dimension $10$ , and the map has zero kernel because $sp_{4}$ is simple and the action on $W$ is non-zero, so it is injective, hence an isomorphism by dimension count. Under this isomorphism the standard $C^{4} = V$ of $sp_{4}$ becomes a four-dimensional irreducible of $so_{5}$ with highest weight $\frac{1}{2} (L_{1} + L_{2})$ — half-integer entries — which is the spin representation, confirming that $C^{4}$ is a spin module on the orthogonal side and explaining why it is invisible to the orthogonal tensor construction. $□$

Connections Master

Schur-Weyl duality 07.05.04. The direct prerequisite and the type- $A$ template. There the centraliser of $GL (V)$ on $V^{\otimes d}$ is the group algebra $C [S_{d}]$ , the Weyl modules $S_{λ} (V)$ are already irreducible, and no contraction is needed because $GL (V)$ preserves no bilinear form. The present unit is the structural sequel: adding an invariant form $B$ enlarges the centraliser to the Brauer algebra $B_{d} (δ)$ , and the irreducibles become the traceless parts $S_{[λ]} (V) = S_{λ} (V) \cap V^{⟨ d ⟩}$ . The Schur functor of the prerequisite is the first factor; the traceless projection is the new second factor.
Representations of $sl_{2} C$ 07.06.11. The rank-one ladder reappears twice. First as the engine of integrality: the highest weight read off a Young diagram is dominant integral because each simple-root $sl_{2}$ -triple sees an integer ladder, exactly as in the prerequisite. Second, and more strikingly, the contraction-and-wedge pair $(C, ω \land)$ of the symplectic Lefschetz decomposition is literally an $sl_{2}$ -triple acting on the exterior algebra, and the primitive (traceless) pieces are its lowest-weight vectors, computed by the same $sl_{2}$ representation theory developed in the prerequisite.
Representations of $sl_{3} C$ 07.06.12. The rank-two worked case generalised. There the two fundamental representations of $sl_{3}$ were the exterior powers $Λ^{1} V$ and $Λ^{2} V$ of the standard representation; here the $n$ fundamental representations of $sp_{2 n}$ are the primitive exterior powers $Λ_{0}^{k} V$ for $1 \leq k \leq n$ , and the fundamentals of $so_{m}$ are the harmonic exterior powers together with the spin representations. The highest-weight-off-a-diagram bookkeeping and the Weyl-dimension computations of the rank-two unit are the special case of the general classical-group pattern this unit constructs.
Spin geometry and Clifford algebras. The deliberate gap. The Weyl tensor construction realises every classical-group irreducible whose highest weight is a partition but never the spin or half-spin representations, whose highest weights $\frac{1}{2} (L_{1} + \dots \pm L_{n})$ have half-integer entries outside every tensor power. Those are built as irreducible modules over the Clifford algebra $Cl (V, B)$ , with $so_{m}$ embedded via the quadratic elements $e_{i} e_{j}$ and the volume element splitting the half-spin pieces for even $m$ — the content of the spin-representation development in the geometry chapters and the source of the four-dimensional standard representation of $sp_{4} ≅ so_{5}$ .

Historical & philosophical context Master

The tensor construction of the classical-group irreducibles is the central achievement of Hermann Weyl's 1939 monograph The Classical Groups: Their Invariants and Representations (Princeton University Press) ^{[Weyl 1939]}, a book that consolidated two decades of work into a single methodological program: study the representations of $GL$ , $O$ , and $Sp$ together with their invariant theory, using the double-commutant (centraliser) method as the unifying tool. Weyl's insight was that the symmetric group's role in the type- $A$ Schur-Weyl duality — itself going back to Issai Schur's 1901 dissertation — had a natural enlargement for the groups preserving a bilinear form, and that the irreducible representations could be cut out of tensor powers by combining Young symmetrisers with the trace-free condition imposed by the form. The book's title deliberately echoed the nineteenth-century invariant-theory tradition of Cayley, Sylvester, and Gordan, recasting their "symbolic method" in the modern language of group representations.

The precise centraliser algebra was identified by Richard Brauer in his 1937 Annals of Mathematics paper On algebras which are connected with the semisimple continuous groups (vol. 38, 857-872) ^{[Brauer 1937]}. Brauer constructed what is now called the Brauer algebra $B_{d} (δ)$ , whose basis consists of perfect matchings (diagrams) of $2 d$ points, with the parameter $δ$ specialised to $m$ for the orthogonal group and to $- 2 n$ for the symplectic group. Brauer proved that this diagram algebra is exactly the centraliser of $O (m)$ or $Sp (2 n)$ on $V^{\otimes d}$ , completing the orthogonal/symplectic analogue of Schur-Weyl duality. The negative parameter $- 2 n$ for the symplectic case — a dimension that is "negative" in the diagrammatic bookkeeping — was an early instance of the super-dimension phenomenon later systematised in the theory of Deligne categories.

The one family of representations the tensor construction cannot reach, the spin representations, was discovered earlier and independently by Élie Cartan in his 1913 paper Les groupes projectifs qui ne laissent invariante aucune multiplicité plane (Bulletin de la Société Mathématique de France 41, 53-96) ^{[Cartan 1913]}. Cartan found the irreducible representations of $so_{m}$ with half-integer highest weights as a structural consequence of the classification of simple Lie algebras, fully twenty-six years before Weyl's monograph, and these spinor representations entered physics through Dirac's 1928 equation, where the four-component spinor is precisely the spin representation of $so_{1, 3}$ . The historical sequence is telling: the abstract highest-weight classification (Cartan) predicted representations that the concrete tensor construction (Weyl, Brauer) could not produce, marking the precise boundary between the tensor and the spinor worlds. Roger Howe's 1989 Transactions of the AMS paper Remarks on classical invariant theory (vol. 313, 539-570) ^{[Howe 1989]} later revealed that the contraction maps of the Weyl construction are the action of a dual partner in a reductive pair, unifying the harmonic decomposition, the theta correspondence, and the classical first fundamental theorems under a single organising principle. The philosophical lesson is that the same family of objects — the irreducible representations of a classical group — admits two complementary descriptions, one combinatorial-tensorial and one abstract-weight-theoretic, and that the places where they fail to coincide (the spin representations) are exactly the places where the deepest geometry lives.

Bibliography Master

@book{Weyl1939Classical,
  author    = {Weyl, Hermann},
  title     = {The Classical Groups: Their Invariants and Representations},
  publisher = {Princeton University Press},
  year      = {1939}
}

@article{Brauer1937,
  author  = {Brauer, Richard},
  title   = {On algebras which are connected with the semisimple continuous groups},
  journal = {Annals of Mathematics},
  volume  = {38},
  year    = {1937},
  pages   = {857--872}
}

@article{Cartan1913Spin,
  author  = {Cartan, {\'E}lie},
  title   = {Les groupes projectifs qui ne laissent invariante aucune multiplicit{\'e} plane},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {41},
  year    = {1913},
  pages   = {53--96}
}

@article{Howe1989,
  author  = {Howe, Roger},
  title   = {Remarks on classical invariant theory},
  journal = {Transactions of the American Mathematical Society},
  volume  = {313},
  year    = {1989},
  pages   = {539--570}
}

@book{FultonHarris,
  author    = {Fulton, William and Harris, Joe},
  title     = {Representation Theory: A First Course},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {129},
  year      = {1991}
}

@book{GoodmanWallach,
  author    = {Goodman, Roe and Wallach, Nolan R.},
  title     = {Symmetry, Representations, and Invariants},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics},
  volume    = {255},
  year      = {2009}
}

@book{Procesi,
  author    = {Procesi, Claudio},
  title     = {{L}ie Groups: An Approach through Invariants and Representations},
  publisher = {Springer},
  series    = {Universitext},
  year      = {2007}
}

@book{HallLie,
  author    = {Hall, Brian C.},
  title     = {{L}ie Groups, {L}ie Algebras, and Representations: An Elementary Introduction},
  publisher = {Springer},
  edition   = {2},
  series    = {Graduate Texts in Mathematics},
  volume    = {222},
  year      = {2015}
}

@article{Schur1901,
  author  = {Schur, Issai},
  title   = {{\"U}ber eine {K}lasse von {M}atrizen, die sich einer gegebenen {M}atrix zuordnen lassen},
  journal = {Dissertation, Berlin},
  year    = {1901}
}

Prerequisites

07.05.04
07.06.11
07.06.12

Tier anchors

beginner: Fulton-Harris *Representation Theory* §16 opening (symmetric and exterior powers as the first irreducibles) and the informal picture of cutting tensors down by a bilinear form; Hall *Lie Groups, Lie Algebras, and Representations* Ch. 7 informal account of $\mathfrak{sp}$ and $\mathfrak{so}$ standard representations
intermediate: Fulton-Harris *Representation Theory* §17 (the symplectic Lie algebras $\mathfrak{sp}_{2n}\mathbb{C}$), §18 (the orthogonal Lie algebras $\mathfrak{so}_m\mathbb{C}$), §19 (spin representations); Goodman-Wallach *Symmetry, Representations, and Invariants* Ch. 10 (tensor and harmonic constructions); Procesi *Lie Groups* Ch. 11
master: Hermann Weyl 1939 *The Classical Groups: Their Invariants and Representations* (Princeton) Ch. III-VI; Richard Brauer 1937 *On algebras which are connected with the semisimple continuous groups* (Ann. of Math. 38, 857-872) — the Brauer-algebra centraliser of $\mathrm{O}(V)$ and $\mathrm{Sp}(V)$ on $V^{\otimes d}$; Cartan 1913 *Les groupes projectifs qui ne laissent invariante aucune multiplicité plane* (Bull. SMF 41) — the spin representations; Fulton-Harris *Representation Theory* §§16-19; Goodman-Wallach Ch. 10; Howe 1989 *Remarks on classical invariant theory* (Trans. AMS 313) — the dual-pair organising principle

References

Fulton-Harris — Representation Theory · §16 Lecture 16 (Symmetric and exterior powers; Weyl's construction); §17 Lecture 17 The symplectic Lie algebras $\mathfrak{sp}_{2n}\mathbb{C}$; §18 Lecture 18 The orthogonal Lie algebras $\mathfrak{so}_m\mathbb{C}$; §19 Lecture 19 Spin representations of $\mathfrak{so}_m\mathbb{C}$
Weyl — The Classical Groups: Their Invariants and Representations · Ch. III (the orthogonal and symplectic groups), Ch. IV (matric algebras and the double-commutant/centraliser method), Ch. V-VI (the irreducible representations of $\mathrm{Sp}$ and $\mathrm{O}$ via traceless tensors); Princeton University Press 1939, second edition 1946
Brauer — On algebras which are connected with the semisimple continuous groups · Ann. of Math. 38 (1937) 857-872; the Brauer algebra $B_d(\pm m)$ as the centraliser of $\mathrm{O}(m)$ (parameter $+m$) and $\mathrm{Sp}(2n)$ (parameter $-2n$) acting on $V^{\otimes d}$, replacing the group algebra of $S_d$ in Schur-Weyl duality
Cartan — Les groupes projectifs qui ne laissent invariante aucune multiplicité plane · Bull. Soc. Math. France 41 (1913) 53-96; the discovery of the spin representations of $\mathfrak{so}_m\mathbb{C}$, the irreducibles with half-integer highest weight not obtained from the tensor (Weyl) construction
Goodman-Wallach — Symmetry, Representations, and Invariants · Ch. 9-10; the tensor construction of $\mathrm{GL}$ Weyl modules, the harmonic decomposition for $\mathrm{O}(V)$, the traceless construction for $\mathrm{Sp}(V)$, and the Brauer-algebra Schur-Weyl duality
Howe — Remarks on classical invariant theory · Trans. Amer. Math. Soc. 313 (1989) 539-570; the reductive dual pair $(\mathrm{Sp}(2n), \mathrm{O}(k))$ and $(\mathrm{O}(m), \mathrm{Sp}(2k))$ organising the contractions and the harmonic decomposition as a $\mathfrak{sl}_2$- or $\mathfrak{sp}$-action commuting with the classical group
Procesi — Lie Groups: An Approach through Invariants and Representations · Ch. 6 (Schur-Weyl), Ch. 11 (the Weyl construction for the classical groups and the first fundamental theorem of invariant theory for $\mathrm{O}$ and $\mathrm{Sp}$)

Estimated time

beginner: 22m
intermediate: 55m
master: 95m