Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions

TL;DR

This paper classifies unitary and calibrated representations of cyclotomic Hecke algebras at roots of unity, showing their equivalence for key parameters, and constructs these modules using BGG resolutions.

Contribution

It provides a combinatorial classification and a cohomological construction of these representations, linking multipartition combinatorics with affine Weyl group actions.

Findings

Unitary and calibrated representations coincide at key parameters.

Classification achieved via multipartition combinatorics and affine Weyl group points.

Modules constructed through BGG resolutions.

Abstract

We relate the classes of unitary and calibrated representations of cyclotomic Hecke algebras and, in particular, we show that for the most important deformation parameters these two classes coincide. We classify these representations in terms of both multipartition combinatorics and as the points in the fundamental alcove under the action of an affine Weyl group. Finally, we cohomologically construct these modules via BGG resolutions.