Schur–Weyl duality

Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. It is named after two pioneers of representation theory of Lie groups, Issai Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary and general linear groups.

Schur–Weyl duality can be proven using the double centralizer theorem.^[1]

Consider the tensor space

${\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \cdots \otimes \mathbb {C} ^{n}}$ with k factors.

The symmetric group S_k on k letters acts on this space (on the left) by permuting the factors,

${\displaystyle \sigma (v_{1}\otimes v_{2}\otimes \cdots \otimes v_{k})=v_{\sigma ^{-1}(1)}\otimes v_{\sigma ^{-1}(2)}\otimes \cdots \otimes v_{\sigma ^{-1}(k)}.}$

The general linear group GL_n of invertible n×n matrices acts on it by the simultaneous matrix multiplication,

${\displaystyle g(v_{1}\otimes v_{2}\otimes \cdots \otimes v_{k})=gv_{1}\otimes gv_{2}\otimes \cdots \otimes gv_{k},\quad g\in {\text{GL}}_{n}.}$

These two actions commute, and in its concrete form, the Schur–Weyl duality asserts that under the joint action of the groups S_k and GL_n, the tensor space decomposes into a direct sum of tensor products of irreducible modules (for these two groups) that actually determine each other,

${\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \cdots \otimes \mathbb {C} ^{n}=\bigoplus _{D}\pi _{k}^{D}\otimes \rho _{n}^{D}.}$

The summands are indexed by the Young diagrams D with k boxes and at most n rows, and representations ${\displaystyle \pi _{k}^{D}}$ of S_k with different D are mutually non-isomorphic, and the same is true for representations ${\displaystyle \rho _{n}^{D}}$ of GL_n.

The abstract form of the Schur–Weyl duality asserts that two algebras of operators on the tensor space generated by the actions of GL_n and S_k are the full mutual centralizers in the algebra of the endomorphisms ${\displaystyle \mathrm {End} _{\mathbb {C} }(\mathbb {C} ^{n}\otimes \mathbb {C} ^{n}\otimes \cdots \otimes \mathbb {C} ^{n}).}$

Suppose that k = 2 and n is greater than one. Then the Schur–Weyl duality is the statement that the space of two-tensors decomposes into symmetric and antisymmetric parts, each of which is an irreducible module for GL_n:

${\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{n}=S^{2}\mathbb {C} ^{n}\oplus \Lambda ^{2}\mathbb {C} ^{n}.}$

The symmetric group S₂ consists of two elements and has two irreducible representations, the trivial representation and the sign representation. The trivial representation of S₂ gives rise to the symmetric tensors, which are invariant (i.e. do not change) under the permutation of the factors, and the sign representation corresponds to the skew-symmetric tensors, which flip the sign.

First consider the following setup:

• G a finite group,
• ${\displaystyle A=\mathbb {C} [G]}$ the group algebra of G,
• ${\displaystyle U}$ a finite-dimensional right A-module, and
• ${\displaystyle B=\operatorname {End} _{G}(U)}$, which acts on U from the left and commutes with the right action of G (or of A). In other words, ${\displaystyle B}$ is the centralizer of ${\displaystyle A}$ in the endomorphism ring ${\displaystyle \operatorname {End} _{\mathbb {C} }(U)}$.

The proof uses two algebraic lemmas.

Proof: Since U is semisimple by Maschke's theorem, there is a decomposition ${\displaystyle U=\bigoplus _{i}U_{i}^{\oplus m_{i}}}$ into simple A-modules. Then ${\displaystyle U\otimes _{A}W=\bigoplus _{i}(U_{i}\otimes _{A}W)^{\oplus m_{i}}}$. Since A is the left regular representation of G, each simple G-module appears in A and we have that ${\displaystyle U_{i}\otimes _{A}W=\mathbb {C} }$ (respectively zero) if and only if ${\displaystyle U_{i},W}$ correspond to the same simple factor of A (respectively otherwise). Hence, we have: ${\displaystyle U\otimes _{A}W=(U_{i_{0}}\otimes _{A}W)^{\oplus m_{i_{0}}}=\mathbb {C} ^{\oplus m_{i_{0}}}.}$ Now, it is easy to see that each nonzero vector in ${\displaystyle \mathbb {C} ^{\oplus m_{i_{0}}}}$ generates the whole space as a B-module and so ${\displaystyle U\otimes _{A}W}$ is simple. (In general, a nonzero module is simple if and only if each of its nonzero cyclic submodule coincides with the module.) ${\displaystyle \square }$

Proof: Let ${\displaystyle W=\operatorname {End} (V)}$. The ${\displaystyle W\hookrightarrow \operatorname {End} (U),w\mapsto w^{d}=d!w\otimes \cdots \otimes w}$. Also, the image of W spans the subspace of symmetric tensors ${\displaystyle \operatorname {Sym} ^{d}(W)}$. Since ${\displaystyle B=\operatorname {Sym} ^{d}(W)}$, the image of ${\displaystyle W}$ spans ${\displaystyle B}$. Since ${\displaystyle \operatorname {GL} (V)}$ is dense in W either in the Euclidean topology or in the Zariski topology, the assertion follows. ${\displaystyle \square }$

The Schur–Weyl duality now follows. We take ${\displaystyle G={\mathfrak {S}}_{d}}$ to be the symmetric group and ${\displaystyle U=V^{\otimes d}}$ the d-th tensor power of a finite-dimensional complex vector space V.

Let ${\displaystyle V^{\lambda }}$ denote the irreducible ${\displaystyle {\mathfrak {S}}_{d}}$-representation corresponding to a partition ${\displaystyle \lambda }$ and ${\displaystyle m_{\lambda }=\dim V^{\lambda }}$. Then by Lemma 1

${\displaystyle S^{\lambda }(V):=V^{\otimes d}\otimes _{{\mathfrak {S}}_{d}}V^{\lambda }}$

is irreducible as a ${\displaystyle \operatorname {GL} (V)}$-module. Moreover, when ${\displaystyle A=\bigoplus _{\lambda }(V^{\lambda })^{\oplus m_{\lambda }}}$ is the left semisimple decomposition, we have:^[4]

${\displaystyle V^{\otimes d}=V^{\otimes d}\otimes _{A}A=\bigoplus _{\lambda }(V^{\otimes d}\otimes _{{\mathfrak {S}}_{d}}V^{\lambda })^{\oplus m_{\lambda }}}$,

which is the semisimple decomposition as a ${\displaystyle \operatorname {GL} (V)}$-module.

The Brauer algebra plays the role of the symmetric group in the generalization of the Schur-Weyl duality to the orthogonal and symplectic groups.

More generally, the partition algebra and its subalgebras give rise to a number of generalizations of the Schur-Weyl duality.

• Partition algebra

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• How to constructively/combinatorially prove Schur-Weyl duality?