Chris Bowman

The super strong linkage principle and other curiosities

We will discuss a “super strong linkage principle” which provides degree-wise upper bounds for graded decomposition numbers of symmetric groups (and more general complex reflection groups).
We will go on to review a new approach to understanding the modular representations of Hecke algebras of complex reflection groups. This approach allows us to construct lots of different graded cellular bases of these Hecke algebras. These new cellular bases allow us to restrict our attention from the whole module category to a subcategory of representations which is both rich in structure, but also understandable over fields of sufficiently large characteristic. We hence provide higher-level generalisations of the familiar “generic behaviour” and Kazhdan–Lusztig theory and settle Martin–Woodcock’s conjecture.