Appears in collection : Combinatorics and Arithmetic for Physics: special days 2023

The dual Cauchy identity is the character version of the $GL_n \times GL_k$ action on the exterior algebra of the natural representation. Additionally, (up to normalization) it is an example of a Schur measure on random partitions. By using other Lie groups (more precisely, dual reductive pairs), we can get analogous representation theoretic statements, which is known as skew Howe duality, and take the corresponding characters. In this talk, we will consider the measure by further specializing the characters to their dimensions to get a probability measure on partitions and describe their limit shapes for a number of dual reductive pairs. This is based on joint work with Anton Nazarov and Olga Postnova.