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The Schur algebra is the commutant of place permutations on a tensor power. It is the linear span of the diagonal general linear group action, by Schur–Weyl duality. Modules over $S(m,n)$ correspond to homogeneous degree-$n$ polynomial representations. Over the complex numbers it is a semisimple algebra: decompose the tensor power as a module for the semisimple group algebra $\mathbb{C}{S}_n$, and take its endomorphism algebra.

For a partition of an integer $\lambda ⊢n$, applying a Young symmetrizer of that shape to the tensor power of each vector space gives a Schur functor. The construction respects linear maps because their tensor powers commute with place permutations. For single rows it gives symmetric powers, and for single columns exterior powers.

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 $\dim V$ 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.

For the unitary group $U(n)$ with $n\ge 3$, the third tensor power of its defining representation decomposes as displayed. The first two summands have highest weights $(3,0,\dots )$ and $(1,1,1,0,\dots )$ and dimensions $\left(\genfrac{}{}{0ex}{}{n+2}{3}\right)$ and $\left(\genfrac{}{}{0ex}{}{n}{3}\right)$. The mixed summand has highest weight $(2,1,0,\dots )$ and dimension $n(n^2-1)/3$. In the three-dimensional weight space spanned by permutations of $e_1\otimes e_1\otimes e_2$, the vectors killed by all positive root vectors have coefficient sum zero, giving multiplicity two. For $n=2$, the exterior cube vanishes and $V_{(2,1)}=\det \otimes {\mathbb{C}}^2$.

The tensor square is the second tensor power $V^{\otimes 2}=V\otimes V$. Over a field of characteristic different from two it decomposes as the direct sum of the symmetric square and exterior square.

For a Young tableau $t$, let $r_t$ sum its row permutations and $c_t$ be the signed sum of its column permutations in the group algebra of the symmetric group. The product $h_t=c_t{r}_t$ is a Young symmetrizer. The other order is another conventional realization. It satisfies $h_t^2=H_{\lambda }{h}_t$, where $H_{\lambda }$ is the hook product of a partition, so division by that scalar gives a primitive idempotent in characteristic zero. Acting on a tensor power constructs a Schur module.