Abstract:
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.
