Complexity Drops for p-by-p Specht Modules

Definitions and mathematical statement

The following formulates the question using the usual modular representation conventions. These conventions, including the meaning of a $p\times p$ partition and the ground field, are not explicitly defined in the supplied record and require confirmation against the report.

Let $p$ be a prime, let $k$ be an algebraically closed field of characteristic $p$, and let $n\geq 1$. Write a partition of $n$ as \[ \lambda=(a_1^{m_1},\ldots,a_t^{m_t}),\qquad a_1>\cdots>a_t>0, \] where $m_j$ is the multiplicity of the part $a_j$. Under the standard convention used here, $\lambda$ is a $p\times p$ partition if \[ p\mid a_j\quad\text{and}\quad p\mid m_j\qquad(1\leq j\leq t). \] Thus its Young diagram consists of $p\times p$ squares arranged according to a partition; the term does not mean only the single square partition $(p^p)$.

Let $\operatorname{core}_p(\lambda)$ be the $p$-core obtained by successively removing rim hooks of length $p$. The block weight of $\lambda$ is \[ w=\frac{n-\lvert\operatorname{core}_p(\lambda)\rvert}{p}. \] Let $S_k^\lambda$ denote the Specht module for the symmetric group $\mathfrak S_n$, constructed over $k$ from the partition $\lambda$. No $p$-regularity assumption on $\lambda$ is imposed.

For a finite-dimensional $k\mathfrak S_n$-module $M$, let \[ \cdots\longrightarrow P_2\longrightarrow P_1\longrightarrow P_0\longrightarrow M\longrightarrow0 \] be a minimal projective resolution. Its complexity $c_{k\mathfrak S_n}(M)$ is zero when $M$ is projective; otherwise it is the least positive integer $c$ for which some constant $C>0$ satisfies \[ \dim_k P_r\leq C r^{c-1} \qquad\text{for all sufficiently large integers }r. \]

Question under these conventions. For every prime $p$, every such field $k$, and every nonempty $p\times p$ partition $\lambda$ of positive block weight $w$, must \[ c_{k\mathfrak S_n}(S_k^\lambda)=w-1 \] hold, or do there exist such partitions for which \[ c_{k\mathfrak S_n}(S_k^\lambda)<w-1? \] Describe these cases further, in particular by characterizing any partitions exhibiting a strict drop and determining their complexities.

The nonempty, positive-weight scope is an explicit interpretation: the supplied question does not discuss the empty partition or weight zero. The source asks about equality versus a smaller value; this draft does not independently establish the upper bound implicit in that framing. The requested description of cases is not given a more specific classification target in the record.

Short English statement

For a $p\times p$ partition $\lambda$ of block weight $w$, ask whether its Specht module always has complexity $w-1$, or whether the complexity can be smaller. Describe any such cases more precisely. The modular setting and partition convention stated above require source confirmation; this is a statement-only draft.

Sources

• Local UnsolvedMath registry: permanent ID $30001192$, title Complexity Drops for p-by-p Specht Modules, exact source problem code catalogue entry; supplied complete source record in the Oberwolfach Reports Open Problems collection.
• Supplied report attribution: Mini-Workshop: Support Varieties (2009), Oberwolfach Reports, pp. 549--592, DOI 10.4171/owr/2009/10. https://doi.org/10.4171/owr/2009/10. The record gives proposed year 2009 and names no proposer. It describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Hemmer 2009. https://doi.org/10.1007/s10801-009-0169-0.
• Supplied literature reference: Lim 2009. https://doi.org/10.1016/j.jalgebra.2009.01.016.
• Supplied literature reference labelled "original Oberwolfach problem": https://doi.org/10.4171/owr/2009/10.
• Supplied literature assessment, checked 2026-08-21: status "partially solved." It reports complexity $p-1$ for the basic $p\times p$ partition and in several special families, while describing the possibility of a lower complexity for arbitrary $p\times p$ partitions of weight $w$ as open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the special families are not specified here.
• Supplied statement-verification metadata: status corrected_verified. The record states that the mathematical content was checked against the official report and that notation, context, and question form were normalized without changing the source-level claim. This draft reports that verification claim without independently confirming it.