Determine the decomposition numbers for symmetric-group Specht modules

Determine the decomposition numbers [S^\lambda:D^\mu] for all partitions \lambda and \mu of n with \mu p-regular in the modular representation theory of the symmetric group.

Background

The paper identifies the decomposition numbers of Specht modules for symmetric groups in characteristic p as a fundamental unresolved problem in modular representation theory. Here, S^\lambda denotes a Specht module and D^\mu denotes the simple module associated with a p-regular partition \mu. Although this problem concerns symmetric groups rather than the centraliser algebras studied in the main results, it provides important representation-theoretic context for the paper.