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

TL;DR

This paper constructs a categorification of the modular data for unipotent characters of complex reflection groups using quantum groups, revealing new structural insights and proving positivity conjectures.

Contribution

It introduces a novel categorification approach for modular data of complex reflection groups via quantum group representations and tensor products.

Findings

Categorification of modular data for $G(d,1,n)$

Representation of quantum $rak{gl}_m$ provides categorical interpretation

Proves positivity conjectures of Cuntz at decategorified level

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 $\mathfrak{gl}_m$, 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.