Fourier matrices for $G(d,1,n)$ from quantum general linear groups

Abstract

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group G(d, 1, n). The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table S of the cyclic group of order d. The representation of the quantum universal enveloping algebra of the general linear Lie algebra gl m , with quantum parameter an even root of unity of order 2d, provides a categorical interpretation of the matrix m S. We also prove some positivity conjectures of Cuntz at the decategorified level.