  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
$\mathfrak{gl}_m$, with quantum parameter an even root of unity of order $2d$,
provides a categorical interpretation of the matrix $\bigwedge^m S$. We also
prove some positivity conjectures of Cuntz at the decategorified level.
