Maximal Complexity of Non-p-by-p Specht Modules

Definitions and mathematical statement

Let $p$ be a prime and let $k$ be a field of characteristic $p$. These are the standard modular representation-theoretic parameters implicit in the supplied statement; the record does not specify the coefficient field or whether it is assumed algebraically closed. The formulation below uses this interpretation and requires confirmation of the intended field hypotheses.

For a partition $\lambda\vdash n$, let $S_k^\lambda$ denote the Specht module for the symmetric group $\mathfrak S_n$ over $k$, constructed by the usual polytabloid construction. Let $\operatorname{core}_p(\lambda)$ be its $p$-core, obtained by successively removing rim hooks of length $p$. The weight of the block containing $S_k^\lambda$ is \[ w=\frac{n-\lvert\operatorname{core}_p(\lambda)\rvert}{p}. \] For each positive integer $a$, write $m_a(\lambda)$ for the multiplicity of the part $a$ in $\lambda$. Under the standard meaning of the terminology, $\lambda$ is $p\times p$ if every occurring part is divisible by $p$ and every part multiplicity is divisible by $p$; explicitly, \[ m_a(\lambda)>0\quad\Longrightarrow\quad p\mid a\ \text{ and }\ p\mid m_a(\lambda). \] This meaning of $p\times p$ is not explicitly defined in the supplied record and should be checked against the cited report.

For a finite-dimensional $k\mathfrak S_n$-module $M$, define its complexity $c_{k\mathfrak S_n}(M)$ to be zero when $M$ is projective, and otherwise the least positive integer $c$ for which the terms $P_d$ of a minimal projective resolution satisfy \[ \dim_k P_d\leq C d^{c-1} \qquad\text{for some }C>0\text{ and all sufficiently large }d. \]

The requested assertion, with the preceding conventions, is: for every prime $p$, every coefficient field $k$ of characteristic $p$ admitted by the intended source hypotheses, every positive integer $n$, and every partition $\lambda\vdash n$, \[ \lambda\text{ is not }p\times p \quad\Longrightarrow\quad c_{k\mathfrak S_n}(S_k^\lambda) =\frac{n-\lvert\operatorname{core}_p(\lambda)\rvert}{p} =w. \] The supplied record describes $w$ as the maximal complexity in this block and identifies this implication as the remaining direction of the VIGRE conjecture. Its phrase "every remaining case" does not specify an exceptional set or a list of cases still requiring proof; the displayed implication expresses the general target, including cases reported as already established. No restriction on the prime or block weight is supplied. If the empty partition is included, it is $p\times p$ under the stated definition, so it contributes no instance of the implication.

The supplied dated assessment labels the problem partially solved and reports established abelian-defect, hook, low-weight, and several structured families. It gives neither precise hypotheses nor ranges for these families. These are supplied literature qualifications, not independently verified results of this draft.

Short English statement

The target is to show that a Specht module whose partition is not $p\times p$ has complexity equal to its block weight $w$, described in the record as the largest possible complexity in that block. The coefficient-field hypotheses and the precise scope of previously proved families require source review. The supplied partially solved status is dated curation.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30001190$, source problem code catalogue entry, title "Maximal Complexity of Non-p-by-p Specht Modules," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2009 and identifies no individual proposer.
• Supplied primary provenance: Mini-Workshop: Support Varieties (2009), Oberwolfach Reports, pp. 549--592, DOI 10.4171/owr/2009/10. https://doi.org/10.4171/owr/2009/10. This is also the supplied literature link labelled "original Oberwolfach problem." No problem-specific page is supplied.
• Supplied literature reference labelled "Hemmer 2009": https://doi.org/10.1007/s10801-009-0169-0.
• Supplied literature reference labelled "Lim 2009": https://doi.org/10.1016/j.jalgebra.2009.01.016.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified here. It labels the problem partially solved, states that the general non-$p\times p$ direction remains conjectural, and reports proved abelian-defect, hook, low-weight, and several structured families. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports that the mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification claim; this draft does not independently verify it or claim to have visited any source.