On an analogue of the James conjecture

TL;DR

This paper presents counterexamples to a proposed analogue of the James conjecture in the context of Khovanov-Lauda-Rouquier algebras, challenging previous optimistic assumptions in representation theory.

Contribution

It provides the first explicit counterexample to the analogue of the James conjecture for simply-laced types, expanding understanding of algebraic structures in representation theory.

Findings

Counterexample in type A_5 for p=2

Counterexamples in all characteristics using recent results

Shows reducibility of characteristic variety in specific cases

Abstract

We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type A_5 for p = 2 and involves the same singularity used by Kashiwara and Saito to show the reducibility of the characteristic variety of an intersection cohomology D-module on a quiver variety. Using recent results of Polo one can give counterexamples in type A in all characteristics.