Let λ be a Young diagram, and e a positive integer. Given a
node (i,j) of λ, we define its residue to be j-i (mod e), and
we define the content of λ to be the multiset consisting of the
residues of all its nodes. Now we say that two partitions lie in the same
e-block if their Young diagrams have the same content. These notions all
come from the modular representation theory of the symmetric group, where
(if e is a prime) the partitions in an e-block are the labels of the
Specht modules lying in an e-block of Sn. It is pretty well known that
two partitions lie in the same e-block if and only if they have the same
e-weight and e-core, and this makes e-blocks much easier to
understand from a combinatorial point of view.
The same set-up applies to multipartitions, and has
representation-theoretic importance. But the appropriate notions of weight
and core for multipartitions are not so easy to come by. In this talk I'll
explain my attempts to define and study these.