James's Conjecture holds for blocks of $q$-Schur algebras of weights 3 and 4

TL;DR

This paper proves that for fields with characteristic at least 5, the adjustment matrix for certain blocks of q-Schur algebras is trivial, and the decomposition numbers are at most one for weight 3 blocks.

Contribution

It establishes that James's Conjecture holds for blocks of q-Schur algebras of weights 3 and 4 in characteristic at least 5, showing the adjustment matrix is the identity.

Findings

Adjustment matrix is the identity for weights 3 and 4

Decomposition numbers for weight 3 blocks are at most one

James's Conjecture holds for these blocks in the specified characteristic

Abstract

When the characteristic of the underlying field is at least 5, we prove that the adjustment matrix for blocks of $q$-Schur algebras of weights 3 and 4 is the identity matrix. Moreover, we show that the decomposition numbers for weight 3 blocks of $q$-Schur algebras are bounded above by one.