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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 symmetric algebra is the quotient of $T(V)$ by the ideal generated by $v\otimes w-w\otimes v$. It is graded by symmetric powers:

The three-dimensional irreducible representation $V_2$ of SU(2) has weights $2,0,-2$. Thus the character of its $n$th symmetric power is ${\sum }_{a+b+c=n}{t}^{2(a-c)}$. Its weight $2\ell$ has multiplicity $\lfloor (n-|\ell |)/2\rfloor +1$ for $|\ell |\le n$. Subtracting adjacent weight multiplicities gives one copy of each $V_{2n},V_{2n-4},\dots$. The invariant subspace is one-dimensional for even $n$ and zero for odd $n$.

The tensor product of Lie algebra representations has action $x(v\otimes w)=xv\otimes w+v\otimes xw$. The Lie algebra representation identity follows because operators on the two different tensor factors commute. This action descends to exterior powers and symmetric powers.

Three identical quadrupole quanta occupy the third symmetric power of the five-dimensional spin-two space. Its dimension is $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$, and its irreducible representations have $J=0,2,3,4,6$, each once. All have positive parity. This counts allowed angular states in the ideal harmonic nuclear quadrupole vibration; anharmonic interactions can split their energies.

In the symmetric power ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of the defining unitary group representation, diagonal torus weights are indexed by nonnegative integer tuples summing to $m$, with one monomial per weight space. Torus averaging extracts a monomial from any nonzero invariant complex subspace. The complexified Lie algebra acts by $E_{ij}=x_i{\partial }_{x_j}$ on the symmetric algebra. Whenever $a_j>0$, the displayed action moves one unit of degree from $j$ to $i$ with nonzero coefficient. Finite transfers connect every degree-$m$ monomial, so the subspace is the entire representation. The case $m=0$ is the one-dimensional trivial representation. This proof needs neither the Weyl character formula nor the Weyl dimension formula.