Pavel Turek, OIST

Title: Balanced columns of decomposition matrices

Abstract: The decomposition matrices describe how the irreducible modules of symmetric groups in characteristic zero decompose in prime characteristic. Understanding these matrices, and in particular, finding a combinatorial description of their entries, is a central open problem in the representation theory of symmetric groups. The main result of the talk is a description of columns of these matrices indexed by ‘d-balanced’ partitions for d=2. It is a consequence of a more general result which describes these columns for any d>1 under some additional assumptions. As a further result, we show that there are many 2-balanced partitions. The key players in the proof of the presented results are Foulkes modules, which are used to construct certain projective modules, and the Jantzen-Schaper formula, which allows us to transfer the algebraic problem into a combinatorial system of equalities, which can then be solved using a new algorithm defined on Young diagrams. This is joint work with Bim Gustavsson, David Hemmer and Stacey Law.

Zoom info can be found on the seminar webpage.