nLab
Schur-Weyl duality

Contents

Contents

Idea

Schur-Weyl duality is a relation between the finite-dimensional irreducible representations of the general linear groups GL(k)GL(k) and of the symmetric group Sym(n)Sym(n).

Concretely, for k,nk, n \in \mathbb{N} consider the k×nk \times n-dimensional complex vector space; regarded as a tensor product of nn copies of n\mathbb{C}^n:

k kntensorfactorsVectorSpaces \underset{ n \; tensor \; factors }{ \underbrace{ \mathbb{C}^k \otimes \cdots \otimes \mathbb{C}^k } } \;\;\; \in \; \mathbb{C}VectorSpaces

As such, this carries

The statement of Schur-Weyl duality is that the decomposition of this linear representation of the direct product group GL(k)×Sym(n)GL(k) \times Sym(n) decomposes as a direct sum of tensor products of irreducible representations for either group, as follows:

k kntensorfactors(λ 1λ k)iλ i=nD (λ)S (λ) \underset{ n \; tensor \; factors }{ \underbrace{ \mathbb{C}^k \otimes \cdots \otimes \mathbb{C}^k } } \;\simeq\; \underset{ { (\lambda_1 \geq \cdots \geq \lambda_k) } \atop { \underset{i}{\sum} \lambda_i = n } }{\oplus} D^{(\lambda)} \otimes S^{(\lambda)}

Here the direct sum is indexed by partitions λ\lambda of nn by kk positive summands, and D (λ)D^{(\lambda)} and S (λ)S^{(\lambda)} denote the irreducible representations of the general linear group and of the symmetric group (Specht modules), respectively, which are indexed by these according to the representation theory of the general linear group and the representation theory of the symmetric group, respectively.

References

Original article:

  • Hermann Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939 (jstor:j.ctv3hh48t)

Review:

Textbook accounts:

In a context of quantum information theory:

  • Jordi Tura i Brugués, Schur-Weyl duality (pdf), Appendix A in: Characterizing Entanglement and Quantum Correlations Constrained by Symmetry, Springer Theses (2017) (doi:10.1007/978-3-319-49571-2)

See also

In the AdS-CFT correspondence:

Last revised on May 20, 2021 at 01:53:35. See the history of this page for a list of all contributions to it.