Containment of Simultaneous Core Partitions

Definitions and mathematical statement

Simultaneous-core containment question. For coprime positive integers $p,q$, is the Young diagram of every partition that is simultaneously a $p$-core and a $q$-core contained in that of the distinguished partition $\kappa_{p,q}$? A partition $\lambda=(\lambda_1\ge\cdots\ge\lambda_\ell>0)$ has diagram $[\lambda]=\{(i,j):1\le j\le\lambda_i\}$. The hook length at $(i,j)$ is $\lambda_i-j+\lambda'_j-i+1$, where $\lambda'$ is the conjugate partition. A $p$-core has no hook length divisible by $p$. Both core conditions are required at once; the source is not asking about the union of the separate $p$-core and $q$-core classes. The parameters are integers, not silently restricted to primes.

With parts padded by zeros, the desired containment is\[\lambda_i\le(\kappa_{p,q})_i\quad\text{for every }i\ge1.\] A bound only on $|\lambda|$, or maximality under another partition order, does not by itself give this coordinatewise diagram containment. The record does not define $\kappa_{p,q}$; its assessment refers to the maximal simultaneous core. Under the usual intended convention, that distinguished partition may be specified by its first-column hook lengths: let $B_{p,q}={\mathbb{Z}}_{>0}\setminus\{ap+bq:a,b\in{\mathbb{Z}}_{\ge0}\}$, list its finitely many elements as $b_1>\cdots>b_r$, and set $(\kappa_{p,q})_i=b_i-r+i$. This is an explicit conventional interpretation of the missing notation, to be checked against the source, not a proof of the containment claim. When $p=1$ or $q=1$, this gap set and the only simultaneous core are empty; a restriction to $p,q\ge2$ would isolate the nontrivial cases but is not given in the record.

The supplied triage checked 2026-08-21 labels the question open. It reports that the report asked the containment question and that later work gives maximal-core results and classifications without establishing the stated containment in general. The exact ordering meant by maximal and the cited results' scope are not reproduced. Those dated claims require independent verification; the original inclusion question is preserved without an independently asserted present-day status.

Short English statement

Does every simultaneous $(p,q)$-core fit row by row inside the distinguished largest core? Diagram containment is stronger than merely having no more boxes. The standard semigroup-gap interpretation of $\kappa_{p,q}$ is explicit but needs source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID 30000448, code catalogue entry; Containment of Simultaneous Core Partitions, Oberwolfach Reports Open Problems.
• Representations of Finite Groups (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/15, pp. 895--978. https://doi.org/10.4171/owr/2006/15. Proposed year and year tag: 2006; no proposer named.
• Supplied literature label OWR report Representations of Finite Groups: https://doi.org/10.4171/owr/2006/15.
• Supplied literature label Olsson Stanton block inclusions and cores: https://doi.org/10.1007/s00010-006-2867-1.
• Literature triage checked 2026-08-21, registry status open; record updated 2026-09-16. The corrected_verified field concerns delimiters only. Verify the dated containment-status assessment independently.