Singular Vectors of the Ding-Iohara-Miki Algebra

Abstract

We review properties of generalized Macdonald functions arising from the AGT correspondence. In particular, we explain a coincidence between generalized Macdonald functions and singular vectors of a certain algebra \({\cal A}(N)\) obtained using the level-(N, 0) representation (horizontal representation) of the Ding-Iohara-Miki algebra. Moreover, we give a factored formula for the Kac determinant of \({\cal A}(N)\), which proves the conjecture that the Poincaré-Birkhoff-Witt-type vectors of the algebra \({\cal A}(N)\) form a basis in its representation space.