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On the Random Young Diagrams and Their Cores

✍ Scribed by Nathan Lulov; Boris Pittel

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
290 KB
Volume
86
Category
Article
ISSN
0097-3165

No coin nor oath required. For personal study only.

✦ Synopsis

An r-core of a Young diagram * is a residual subdiagram obtained after consecutive removals of the feasible r-long border strips, ``rim hooks.'' The removal process on the diagram * and the resulting r-core are the essential elements in the Murnaghan Nakayama formula for / * , the character of the associated irreducible representation of S n (n= |*| ), on the conjugacy class [r [nΓ‚r] ] (n#0 mod r). A complete characterization of r-cores is given, which extends a well known result for r=2. Under an assumption that the partition * is chosen uniformly at random out of all partitions of n, it is shown that typically the r-core size is of order n 1Γ‚2 , while the height and the width are of order n 1Γ‚4 . For n chosen uniformly at random between 1 and N, the core boundary scaled by N 1Γ‚4 is proved to converge, in distribution, to a random concave curve which consists of r&1 line segments.

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