On the random Young diagrams and their cores

An r-core of a Young diagram lambda is a residual subdiagram obtained after consecutive removals of the feasible I-long border strips, "rim hooks." Th e removal process on the diagram lambda, and the resulting I-core are the e ssential elements in the Murnaghan-Nakayama formula for chi(lambda), the ch aracter of the associated irreducible representation of S-n (n = \lambda\), on the conjugacy class {r([n/r])} (n equivalent to 0 mod r). A complete ch aracterization of r-cores is given, which extends a well known result for r =2. Under an assumption that the partition lambda is chosen uniformly at r andom out of all partitions of,l, it is shown that typically the r-core siz e is of order n(1/2), while the height and the width are of order n(1/4). F or n chosen uniformly at random between 1 and N the core boundary scaled by N-1/4 is proved to converge, in distribution, to a random concave curve wh ich consists of r-1 line segments. (C) 1999 Academic Press.