Abstract
The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores — partitions which are both $s$- and $t$-cores %mdash; playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang.In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525—1539], studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core.
Highlights
The study of integer partitions is a very active subject, with connections to representation theory, number theory and symmetric function theory
A prominent theme is the study of s-core partitions, when s is a natural number: we say that a partition λ is an score if it does not have a rim hook of length s; if λ is any partition, the s-core of λ is the partition obtained by repeatedly removing rim s-hooks
In the case where s is a prime, s-cores play an important role in the s-modular representation theory of the symmetric group
Summary
The study of integer partitions is a very active subject, with connections to representation theory, number theory and symmetric function theory. We connect Armstrong’s conjectures to the level t action of the affine symmetric group on the set of s-cores, and present variants of these conjectures, in which the size of an ps, tq-core λ is weighted by the reciprocal of the order of the stabiliser of λ under this action. These weighted averages are (apparently) given by simple formulæ which are very similar to those in Armstrong’s conjectures. We remark that since this paper first appeared in preprint form, our main conjectures have been proved by Wang [W]
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