Modular decomposition numbers of cyclotomic Hecke and diagrammatic Cherednik algebras: A path theoretic approach

TL;DR

This paper develops a path-theoretic framework to analyze the representation theory of cyclotomic Hecke and Cherednik algebras, providing new bounds on decomposition numbers and insights into higher-level Lusztig conjectures.

Contribution

It introduces a super-strong linkage principle and generalizes homomorphism notions, advancing understanding of modular representations of these algebras.

Findings

Degree-wise upper bounds for graded decomposition numbers

Generalized homomorphisms between modules in alcove geometries

Evidence for a higher-level Lusztig conjecture in large characteristic

Abstract

We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a "super-strong linkage principle" which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalise the notion of homomorphisms between Weyl/Specht modules which are "generically" placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.