Let $G=SO(2r)$ and let $\{e_i\pm e_j\}_{i<j\leq r}$ be the positive roots and so the coroots are given by $\check{\alpha}=\frac{2\langle x,\alpha \rangle}{\langle \alpha, \alpha \rangle}$ where $x=(x_1,\ldots,x_r)$. The dimension of $V_\lambda$, an irreducible representation of $G$ with highest weight $\lambda$ is given by $$\frac{\prod \check{\alpha}(\lambda + \rho)}{\prod \check{\alpha}(\rho)}$$ where $\rho$ is the half-sum of positive roots: $(r-1,r-2,\ldots,1,0)$. If we consider $\lambda$ partitioning some $j$ and $j\to \infty$ then the numerator looks just like $\prod \check{\alpha}(\lambda)$ which, written as a polynomial in $\lambda_1,\ldots,\lambda_r$ is $\Delta \cdot s_\rho$ where $\Delta = \prod x_i-x_j$ and $s_\rho$ is the Schur polynomial indexed by $\rho$.
This seems to be the same phenomenon for $SO(2n+1)$.
Has this been observed in the literature? Can anybody provide some insight on why this happens?