geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Schur-Weyl 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 $\mathbb{C}^n$:
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:
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.
Original article:
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:
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.