Schur-Weyl Duality

Schur-Weyl duality establishes a fundamental connection between representations of symmetric groups and general linear groups, revealing deep structural relationships.

TheoremSchur-Weyl Duality

Let $V = \mathbb{C}^n$ and consider the tensor power $V^{\otimes k}$ . The symmetric group $S_k$ acts by permuting factors, and $GL_n(\mathbb{C})$ acts diagonally. These actions commute, and:

$\text{End}_{GL_n}(V^{\otimes k}) \cong \mathbb{C}[S_k]$ $\text{End}_{S_k}(V^{\otimes k}) \cong U(\mathfrak{gl}_n)$

Moreover, as a $(\mathfrak{gl}_n, S_k)$ -bimodule: $V^{\otimes k} \cong \bigoplus_{\lambda} V_n(\lambda) \otimes S^\lambda$ where the sum is over partitions $\lambda \vdash k$ with $\ell(\lambda) \leq n$ , $V_n(\lambda)$ are irreducible $\mathfrak{gl}_n$ -modules, and $S^\lambda$ are Specht modules (irreducible $S_k$ -representations).

Interpretation: The irreducible $\mathfrak{gl}_n$ -representations appearing in $V^{\otimes k}$ are indexed by the same partitions that index irreducible $S_k$ -representations, establishing a dictionary between the two theories.

ExampleSchur-Weyl for $k=2$

For $V^{\otimes 2} = V \otimes V$ :

$S_2$ acts via swapping: $(v_1 \otimes v_2) \mapsto (v_2 \otimes v_1)$
Decomposes as $V^{\otimes 2} = \text{Sym}^2(V) \oplus \wedge^2(V)$ (symmetric and alternating tensors)

The symmetric group $S_2$ has irreducibles: trivial and sign

Schur-Weyl duality gives: $V^{\otimes 2} = V_n((2)) \otimes S^{(2)} \,\oplus\, V_n((1,1)) \otimes S^{(1,1)}$ where $\text{Sym}^2(V) \leftrightarrow (2)$ and $\wedge^2(V) \leftrightarrow (1,1)$ .

DefinitionSchur Functors

The Schur functors $\mathbb{S}_\lambda$ send vector spaces to vector spaces: $\mathbb{S}_\lambda(V) = \text{image of } c_T: V^{\otimes k} \to V^{\otimes k}$ where $c_T$ is the Young symmetrizer for partition $\lambda \vdash k$ .

These give the irreducible polynomial representations of $GL_n$ : $V_n(\lambda) = \mathbb{S}_\lambda(\mathbb{C}^n)$ .

TheoremApplications

Representation stability: As $n \to \infty$ , the representations $V_n(\lambda)$ stabilize, leading to representation theory of the infinite symmetric group.

Schur polynomials: Characters of $V_n(\lambda)$ are Schur polynomials $s_\lambda(x_1, \ldots, x_n)$ , connecting to symmetric function theory.

Plethysm: Compositions $\mathbb{S}_\lambda \circ \mathbb{S}_\mu$ encode deep combinatorics (Kronecker coefficients, plethysm coefficients).

Remark

Schur-Weyl duality generalizes to:

Brauer algebras: Orthogonal and symplectic groups
BMW algebras: Quantum groups and knot invariants
Deligne categories: Interpolation to non-integer $n$
Higher categories: Categorification and Khovanov homology

This duality is fundamental in:

Quantum information theory (entanglement, quantum marginal problem)
Algebraic complexity theory (geometric complexity theory program)
Representation stability in topology and algebraic geometry