nLab Schur-Weyl duality

Idea

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

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

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

As such, this carries

• a canonical linear group action by $GL(k) \coloneqq GL(k,\mathbb{C})$, namely the $n$-fold diagonal action of its defining action on $\mathbb{C}^k$ (its fundamental representation);

• a canonical linear group action by $Sym(n)$ given by permutation of the $n$ tensor factors.

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

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

Here the direct sum is indexed by partitions $\lambda$ of $n$ by $k$ positive summands, and $D^{(\lambda)}$ and $S^{(\lambda)}$ 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.

Related concepts

• representation theory of the general linear group

• representation theory of the symmetric group

• Schur-Weyl measure

• Howe duality

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:

• William Fulton, Joe Harris, Section 6.1 of: Representation Theory: a First Course, Springer, Berlin, 1991 (doi:10.1007/978-1-4612-0979-9)

• Ambar N. Sengupta, Section 10.2 of: Representing Finite Groups – A Semisimple Introduction, Springer 2012 (doi:10.1007/978-1-4614-1231-1)

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

• Wikipedia, Schur-Weyl duality

In the AdS-CFT correspondence:

• Sanjaye Ramgoolam, Schur-Weyl duality as an instrument of Gauge-String duality, AIPConf. Proc. 1031:255-265, 2008 (arXiv:0804.2764)

A general mechanism behind the similar dualities is discussed in

• Georgia Benkart, Commuting actions – A tale of two groups, in: Lie Algebras and Their Representations, Contemp. Math. 194 (1996) 1–46

• S. Doty, New versions of Schur-Weyl duality, Finite groups 2003, Walter de Gruyter (2004) 59–71