# The Dipper-Du Conjecture Revisited

## TL;DR

This paper revisits the Dipper-Du Conjecture within the context of rational Cherednik algebras, establishing an analogous result for symmetric group Hecke algebras and providing a new proof over complex numbers.

## Contribution

It extends the Dipper-Du Conjecture to the setting of rational Cherednik algebras and offers a novel proof over the complex field.

## Key findings

- Proved the analogue of the Dipper-Du Conjecture for Hecke algebras in the Cherednik algebra setting.
- Derived a new proof of the Dipper-Du Conjecture over c.
- Connected local representation theory notions to category al of rational Cherednik algebras.

## Abstract

We consider vertices, a notion originating in local representation theory of finite groups, for the category $\mathcal{O}$ of a rational Cherednik algebra and prove the analogue of the Dipper-Du Conjecture for Hecke algebras of symmetric groups in that setting. As a corollary we obtain a new proof of the Dipper-Du Conjecture over $\mathbb{C}$.