Unitary representations of the Cherednik algebra: $V^*$-homology

TL;DR

This paper provides a combinatorial formula for the homology of unitary representations of the cyclotomic rational Cherednik algebra, linking it to Littlewood-Richardson numbers and graded Betti numbers of certain subspace arrangements.

Contribution

It introduces a novel non-negative combinatorial formula for homology and Betti numbers in the context of Cherednik algebra representations.

Findings

Derived a combinatorial formula using Littlewood-Richardson numbers

Connected homology of Cherednik algebra representations to subspace arrangement invariants

Provided explicit calculations for graded Betti numbers

Abstract

We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups.