Multiplicity-free symmetric-algebra model for polynomial GL representations ($\mathrm{Sym}(V\oplus {\Lambda }^{2}V)={⨁}_{\lambda }{D}_{\lambda }(V)$)

The symmetric algebra on $V\oplus {\Lambda }^{2}V$ contains every irreducible polynomial representation of $GL(V)$ exactly once. Its torus character is ${\prod }_i(1-x_i{)}^{-1}{\prod }_{i<j}(1-x_i{x}_j{)}^{-1}$, 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.