Schur–Weyl duality is a fundamental result in representation theory that describes a deep relationship between two types of algebraic structures: the symmetric groups and the general linear groups. Specifically, it provides a duality between representations of the symmetric group \( S_n \) and representations of the general linear group \( GL(V) \) (where \( V \) is a finite-dimensional vector space) for a fixed \( n \).
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The commuting actions of the symmetric group and the general linear group on a tensor power are mutual commutants. The displayed sum ranges over partitions with at most rows. The Specht modules and Schur modules are the simple factors for the two actions. In particular a primitive Young symmetrizer selects one copy of the matching Schur module.