Skew Howe duality and limit shapes for Young diagrams

Abstract

Consider the exterior algebra of the tensor product of two complex vector spaces of dimension n and k. This space could be regarded as a bimodule for the action of dual pairs of Lie groups. For example, for GL(n) x GL(k) - case this exterior algebra decomposes into direct sum of bimodules parametrised by conjugate partitions inside the n x k rectangle. This is the skew Howe duality. On the level of characters the skew Howe duality yields the dual Cauchy identity for the Schur functions. We interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. This combinatorial approach also allows to obtain product formulas for the q-deformations of multiplicities or different dual pairs of Lie groups . We consider the corresponding probability measures on Young diagrams and prove the uniform convergence to the limit shape of Young diagrams in the limit when n and k go to infinity. (Joint work with A.Nazarov and T.Schrimshaw.)