Schur–Weyl duality explains why general linear groups and symmetric groups appear together whenever a vector space is tensored with itself many times. GL(V) acts diagonally on V^{⊗r}; the symmetric group S_r permutes tensor positions. The two actions commute, and—under the classical hypotheses—each is the full commutant of the other.
The theorem is not merely a convenient simultaneous decomposition. It is a double-centralizer statement. One symmetry controls changes inside each tensor factor. The other controls how the tensor factors are rearranged. Their mutual commutants force tensor power to split into paired irreducible representations indexed by the same partitions.
This article connects the BTT routes through Symmetric Group Representations and Young Tableaux, Tensor Products and Duality, and Tensor-Product Multiplicities.
Two commuting actions · Double centralizers · Decomposition · Worked examples · Schur functors · Stable range · Practice
Why tensor powers create two natural symmetries
Let V be an n-dimensional complex vector space. The r-fold tensor power is V^{⊗r}=V⊗···⊗V.
An element g∈GL(V) acts diagonally:
g·(v₁⊗···⊗v_r)=gv₁⊗···⊗gv_r.
A permutation σ∈S_r acts by permuting tensor positions. Fix the convention
σ·(v₁⊗···⊗v_r)=v_{σ^{-1}(1)}⊗···⊗v_{σ^{-1}(r)}.
The inverse makes this a left group action. Other conventions are equivalent but should not be mixed mid-calculation.
The actions commute
Apply g and σ in either order. If g acts first, every factor v_i becomes gv_i, then σ rearranges them. If σ acts first, the factors are rearranged, then g acts identically on every new position.
Therefore
gσ=σg
as operators on V^{⊗r}.
This elementary identity is the doorway to the whole theorem. Commuting actions mean each image lies inside the commutant of the other.
The commutant
If A is a collection of linear operators on a vector space W, its commutant is
End_A(W)={T∈End(W): Ta=aT for every a∈A}.
Thus the symmetric-group action lands in End_{GL(V)}(V^{⊗r}), while the GL(V) action lands in End_{S_r}(V^{⊗r}).
Schur–Weyl duality says these inclusions become equalities in the classical complex setting, after interpreting the GL side as the algebra generated by its image.
The double-centralizer statement
Let ρ denote the GL(V) action and π the group-algebra action of C[S_r]. Then
End_{GL(V)}(V^{⊗r})=π(C[S_r])
and
End_{S_r}(V^{⊗r})=Alg(ρ(GL(V))).
When n≥r, the S_r representation on tensor space is faithful, so the first commutant is literally isomorphic to C[S_r]. When n<r, the group-algebra map has a kernel because antisymmetrizing more than n tensor factors gives zero.
This stable-range distinction is essential. “The centralizer is C[S_r]” is accurate only when the permutation action is faithful.
A first obstruction: exterior powers
If dim V=n, then Λ^{n+1}V=0. The total antisymmetrizer on n+1 tensor positions therefore annihilates V^{⊗(n+1)}.
Thus the sign idempotent corresponding to a column partition of length n+1 lies in the kernel of the symmetric-group action.
This is why only partitions with at most n rows survive in the Schur–Weyl decomposition.
The decomposition
Schur–Weyl duality gives
V^{⊗r}≈⊕_{λ⊢r, ℓ(λ)≤n} S_λ(V)⊗Sp^λ.
Here:
- λ runs over partitions of r with at most n parts;
- S_λ(V) is the irreducible polynomial GL(V) representation of highest weight λ;
- Sp^λ is the Specht module of S_r.
Each pair appears once as a module for the product GL(V)×S_r. Multiplicity on one side becomes dimension on the other.
For example, as a GL(V)-module alone, S_λ(V) appears dim Sp^λ times. As an S_r-module alone, Sp^λ appears dim S_λ(V) times.
Why the same partition labels both sides
Partitions classify irreducible representations of S_r over C. They also classify polynomial irreducible GL_n representations of homogeneous degree r, subject to the row bound ℓ(λ)≤n.
Schur–Weyl duality does not merely notice this shared indexing. It constructs the simultaneous decomposition in which those two λ-labelled objects are paired.
The pairing is the prototype for many later dualities involving centralizer algebras.
Worked example: r=2
The partitions of 2 are (2) and (1,1). Therefore
V⊗V≈Sym²V⊗triv ⊕ Λ²V⊗sgn.
The transposition (12) acts by +1 on symmetric tensors and −1 on alternating tensors.
For n=2, dim Sym²V=3 and dim Λ²V=1. Thus total dimension is 3+1=4=2².
Worked example: n=2, r=3
The partitions of 3 are (3), (2,1) and (1,1,1). The last has three rows and therefore vanishes for GL₂.
Hence
(C²)^{⊗3}≈S_{(3)}(C²)⊗Sp^{(3)} ⊕ S_{(2,1)}(C²)⊗Sp^{(2,1)}.
The dimensions are dim S_{(3)}(C²)=4, dim Sp^{(3)}=1, dim S_{(2,1)}(C²)=2 and dim Sp^{(2,1)}=2.
Therefore
4·1+2·2=8=2³.
The missing sign representation reflects Λ³(C²)=0.
Recover multiplicities through Hom spaces
The GL(V) multiplicity space of S_λ(V) in V^{⊗r} is naturally Sp^λ:
Hom_{GL(V)}(S_λ(V),V^{⊗r})≈Sp^λ.
Likewise,
Hom_{S_r}(Sp^λ,V^{⊗r})≈S_λ(V).
This is the cleanest expression of “multiplicity space becomes a representation of the commuting algebra”. The multiplicity is no longer only a number; it carries the other symmetry.
Schur functors
The construction V↦S_λ(V) is functorial. A linear map f:V→W induces S_λ(f):S_λ(V)→S_λ(W).
Special partitions recover familiar operations:
- λ=(r): S_λ(V)=Sym^rV;
- λ=(1^r): S_λ(V)=Λ^rV;
- general λ: mixed symmetry prescribed by rows and columns of the Young diagram.
Young symmetrizers in C[S_r] project tensor space toward these mixed-symmetry components, though normalization and non-orthogonality require care.
Young symmetrizers
Choose a Young tableau T of shape λ. Let a_T sum permutations preserving rows and b_T alternate permutations preserving columns. A Young symmetrizer has the form
c_T=a_T b_T
or the reverse product, depending on convention.
Applying c_T to V^{⊗r} imposes row symmetries and column antisymmetries. The resulting image realises a copy related to S_λ(V).
The operator c_T is generally only quasi-idempotent before scaling. Treating it as an orthogonal projection without normalization is incorrect.
Schur algebra
The commutant
S(n,r)=End_{C[S_r]}((C^n)^{⊗r})
is the Schur algebra.
Polynomial representations of GL_n homogeneous of degree r can be studied through modules over S(n,r). This finite-dimensional algebra packages the GL_n side of Schur–Weyl duality.
The name Schur algebra should not be confused with the Schur complement in matrix algebra or with Schur functions, although all arise from Issai Schur’s work and connect within representation theory.
Double-centralizer philosophy
Suppose two semisimple algebras A and B act on W and centralize one another. Then W often decomposes as
W≈⊕_i U_i⊗M_i,
with U_i irreducible for A and M_i irreducible for B.
Schur–Weyl duality is the most famous example. The next article replaces the symmetric group by Brauer and Temperley–Lieb diagram algebras when the group preserves a bilinear form or when the tensor category is deformed.
Stable range
When n grows while r is fixed, eventually every partition λ⊢r satisfies ℓ(λ)≤n. Then no partition is excluded by dimension.
This is a basic instance of representation stability: the combinatorial indexing stops changing once the ambient dimension is large enough compared with tensor degree.
Later in Batch 09, FI-modules systematise this idea across sequences of symmetric-group representations.
Quantum Schur–Weyl duality
Replacing GL_n by the quantum group U_q(gl_n) and C[S_r] by the Hecke algebra H_r(q) gives quantum Schur–Weyl duality under suitable generic parameters.
At q=1, the Hecke algebra specialises to the symmetric-group algebra and the quantum group specialises toward the ordinary enveloping algebra picture.
At roots of unity, semisimplicity can fail and the clean decomposition needs modular/tilting refinements. The generic theorem should not be transferred unchanged into singular parameters.
A verification workflow
- 1. Fix n and r.
- 2. State the permutation-action convention.
- 3. Verify commuting actions directly on a pure tensor.
- 4. List partitions λ⊢r.
- 5. Remove those with more than n rows.
- 6. Compute Specht dimensions by hook length.
- 7. Compute GL_n dimensions by Weyl or hook-content formula.
- 8. Check Σ dim S_λ(V)·dim Sp^λ=n^r.
- 9. Distinguish group algebra from its image when n<r.
- 10. Interpret multiplicity spaces as modules for the commuting algebra.
Common mistakes
- Forgetting the inverse in a left permutation action.
- Claiming C[S_r] acts faithfully for every n.
- Keeping partitions with too many rows.
- Confusing multiplicity with representation dimension.
- Treating a Young symmetrizer as an automatically normalized orthogonal projector.
- Assuming double centralizers without semisimplicity or finite-dimensional hypotheses in unrelated settings.
- Mixing Schur algebra, Schur functor and Schur function.
- Applying generic quantum Schur–Weyl statements unchanged at roots of unity.
Practice questions
1. What are the two commuting actions on V^{⊗r}? 2. Why do they commute? 3. Which partitions of 4 survive for GL₂? 4. Decompose V⊗V.
5. Why does Sp^{(1,1,1)} disappear from (C²)^{⊗3}? 6. Check the dimension 8 in the n=2,r=3 example. 7. What is the multiplicity space Hom_{GL(V)}(S_λ(V),V^{⊗r})? 8. What is S(n,r)?
9. What changes when n≥r? 10. Which partition gives Sym^rV? 11. Which gives Λ^rV? 12. What algebra replaces C[S_r] in generic quantum Schur–Weyl duality?
Worked answers
1. GL(V) acts diagonally and S_r permutes tensor positions.
2. Applying the same linear map to every factor is independent of the order in which the factors are rearranged.
3. (4), (3,1) and (2,2); partitions with three or four rows are excluded.
4. Sym²V⊕Λ²V.
5. It corresponds to Λ³(C²)=0.
6. 4·1+2·2=8.
7. The Specht module Sp^λ.
8. End_{C[S_r]}((C^n)^{⊗r}), the Schur algebra.
9. Every partition of r has at most n rows, and the symmetric-group action is faithful.
10. (r).
11. (1^r).
12. The Hecke algebra H_r(q).
Sources and further study
Standard references include William Fulton and Joe Harris, Representation Theory; Bruce Sagan, The Symmetric Group; and Roger Goodman and Nolan Wallach, Symmetry, Representations, and Invariants. The BTT guides on Young Tableaux and Littlewood–Richardson multiplicities supply the combinatorial prerequisites.
Representation Mathematics — Batch 09
Replace symmetric-group centralizers by diagram algebras in Brauer and Temperley–Lieb Algebras. Build quantum groups from quiver counting in Hall Algebras. Study sequences of symmetric-group representations in Representation Stability and FI-Modules. Return to the BTT Mathematics Learning Hub.