Some combinatorial results involving Young diagrams

Source abstract

In the first half of this paper we introduce a new method of examining the q -hook structure of a Young diagram, and use it to prove most of the standard results about q -cores and q -quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q -weight and says how many of them are q -regular. In the case where q is prime, this tells us how many ordinary and q -modular irreducible representations of the symmetric group there are in a given q -block. None of the results of section 2 is original. In the next section we give a new definition, the p -power diagram, which is closely connected with the p -quotient. This concept is interesting because when p is prime a condition involving the p -power diagram appears to be a necessary and sufficient criterion for the diagram to be p -regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p . In this paper we derive combinatorial results involving the p -power diagram, and in a later article we investigate the relevant representation theory.