A counter-example to Martino's conjecture about generic Calogero-Moser families

TL;DR

This paper provides a counter-example to Martino's conjecture by explicitly determining the generic Calogero-Moser families for nine exceptional groups, showing the conjecture fails for G25, the first known exception.

Contribution

The paper explicitly computes the generic Calogero-Moser families for nine exceptional groups and identifies G25 as the only counter-example to Martino's conjecture.

Findings

Martino's conjecture holds for most exceptional groups

G25 is the first known counter-example

Explicit computations confirm the counter-example

Abstract

The Calogero-Moser families are partitions of the irreducible characters of a complex reflection group derived from the block structure of the corresponding restricted rational Cherednik algebra. It was conjectured by Martino in 2009 that the generic Calogero-Moser families coincide with the generic Rouquier families, which are derived from the corresponding Hecke algebra. This conjecture is already proven for the whole infinite series G(m,p,n) and for the exceptional group G4. A combination of theoretical facts with explicit computations enables us to determine the generic Calogero-Moser families for the nine exceptional groups G4, G5, G6, G8, G10, G23=H3, G24, G25, and G26. We show that the conjecture holds for all these groups - except surprisingly for the group G25, thus being the first and only-known counter-example so far.