Classification and dimensions of simple modules of symmetric groups in characteristic p

Determine a complete classification and explicit dimension formulas for the simple (irreducible) representations of symmetric groups over a field of characteristic p > 0.

Background

The authors highlight that, unlike the semisimple situation over characteristic zero, modular representation theory remains poorly understood in general. Even for symmetric groups, fundamental questions such as classifying simple modules and computing their dimensions are not resolved.

This underscores the breadth of open problems in modular representation theory, motivating structural approaches like Broué’s conjectures and Rickard’s splendid equivalences.