Skew Howe duality and limit shapes of Young diagrams

We consider the skew Howe duality for the action of certain dual pairs of Lie
groups $(G_1, G_2)$ on the exterior algebra $\bigwedge(\mathbb{C}^{n} \otimes
\mathbb{C}^{k})$ as a probability measure on Young diagrams by the
decomposition into the sum of irreducible representations. We prove a
combinatorial version of this skew Howe for the pairs $(\mathrm{GL}_{n},
\mathrm{GL}_{k})$, $(\mathrm{SO}_{2n+1}, \mathrm{Pin}_{2k})$,
$(\mathrm{Sp}_{2n}, \mathrm{Sp}_{2k})$, and $(\mathrm{Or}_{2n},
\mathrm{SO}_{k})$ using crystal bases, which allows us to interpret the skew
Howe duality as a natural consequence of lattice paths on lozenge tilings of
certain partial hexagonal domains. The $G_1$-representation multiplicity is
given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and
as a product formula using Dodgson condensation. These admit natural
$q$-analogs that we show equals the $q$-dimension of a $G_2$-representation (up
to an overall factor of $q$), giving a refined version of the combinatorial
skew Howe duality. Using these product formulas (at $q =1$), we take the
infinite rank limit and prove the diagrams converge uniformly to the limit
shape.