Macdonald symmetric functions of rectangular shapes

Using vertex operators we study Macdonald symmetric functions of rectangular shapes and their connection with the q-Dyson Laurent polynomial. We find a vertex operator realization of Macdonald functions and give a generalized Frobenius formula for them. As byproducts of the realization, we find a q-Dyson constant term orthogonality relation which generalizes a conjecture due to Kadell (2000), and we generalize Matsumoto's hyperdeterminant formula for rectangular Jack functions to Macdonald functions.