Skew Howe duality and limit shape of Young diagrams

We consider skew Howe duality that is related to the action of a pair of Lie groups on the exterior algebra $\bigwedge(\mathbb{C}^{n}\otimes\mathbb{C}^{k})$. This exterior algebra admits a multiplicity-free decomposition into a direct sum of tensor products of representations for the pairs of dual groups $(GL_{n},GL_{k})$, $(SO_n,Pin_k)$ and $(Sp_n,Sp_k)$. From the point of view of a single group in a pair this is a tensor power decomposition into the sum of the irreducible representations. Such a decomposition can be used to introduce a probability measure on Young diagrams parameterizing the representations that appear in the sum. In the limit of infinite rank of the groups the diagrams converge to a limit shape. In order to derive the limit shape we need to express the dimension of the dual group representation in terms of the original diagram and thus obtain the explicit formula for the multiplicities of tensor power decomposition into the sum of the irreducibles. We connect this multiplicity to the number of paths in the lozenge tiling of a certain hexagonal domain and use the Lindström–Gessel–Viennot lemma, as well as a recursion of the determinants to prove the product formulas for the multiplicities. These paths are naturally connected to the crystals for the corresponding representations. We then compute the asymptotic of the probability measure and derive the explicit formula for the limit shapes of Young diagrams, that is different from the results of A.M. Vershik, S.V. Kerov and P. Biane. We discover that the limit shapes of Young diagrams for the symplectic and orthogonal groups are “halves” of the limit shape of the general linear group.