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.
If has cycles of length , contraction of matrix entries around its cycles gives . The formula holds for every endomorphism, not only diagonalizable ones. The Schur–Weyl duality decomposition equates it with a sum of products of Specht and Schur characters.
The standard-tableau right-ideal decomposition of , tensored over that algebra with , gives this direct sum of general linear group modules. With , the map is an isomorphism. Individual summands generally need not be symmetric group submodules; instead each is a Schur module for the commuting action.
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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 \( n \).