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.

Schur–Weyl duality can be proven using the double centralizer theorem.

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

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

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

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 determine each other,

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.

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