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. 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.

Description

Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space

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

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

\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 GLn of invertible n×n matrices acts on it by the simultaneous matrix multiplication,

g(v_1\otimes v_2\otimes\cdots\otimes v_k) = gv_1\otimes gv_2\otimes\cdots\otimes gv_k, \quad g\in GL_n.

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

\mathbb{C}^n\otimes\mathbb{C}^n\otimes\cdots\otimes\mathbb{C}^n = \sum_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 \pi_k^D of Sk with different D are mutually non-isomorphic, and the same is true for representations \rho_n^D of GLn.

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

Example

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 GLn:

\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.

References