On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$

TL;DR

This paper proposes and partially proves a conjecture linking two sets of characters for the complex reflection group G(d,1,n), involving Calogero-Moser cells and quantum group representations.

Contribution

It introduces a conjecture connecting Calogero-Moser cell characters with quantum group representations and proves it for specific cases.

Findings

Proved the conjecture for G(d,1,2)

Established the conjecture for generic parameters in G(d,1,n)

Linked complex reflection group characters with quantum algebra representations

Abstract

We propose a conjecture relating two different sets of characters for the complex reflection group $G(d,1,n)$. From one side, the characters are afforded by Calogero-Moser cells, a conjectural generalisation of Kazhdan-Lusztig cells for a complex reflection group. From the other side, the characters arise from a level $d$ irreducible integrable representations of $\mathcal{U}_q(\mathfrak{sl}_{\infty})$. We prove this conjecture in some cases: in full generality for $G(d,1,2)$ and for generic parameters for $G(d,1,n)$.