= Multiplicity-free symmetric-algebra model for polynomial GL representations
{title2=$\operatorname{Sym}(V\oplus\Lambda^2V)=\bigoplus_\lambda D_\lambda(V)$}
The <symmetric algebra> on $V\oplus\Lambda^2V$ contains every irreducible <polynomial representation> of $GL(V)$ exactly once. Its torus <character> is $\prod_i(1-x_i)^{-1}\prod_{i<j}(1-x_ix_j)^{-1}$, which equals $\sum_\lambda s_\lambda(x)$ by the Schur identity. Complete reducibility and linear independence of Schur <characters> identify the summands. This is a formal graded identity: each fixed scalar degree is finite, so no analytic convergence is needed. Multiplicity one refers to irreducible <modules>, not arbitrary reducible representations.
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