Socles of Products from Lusztig Components

Definitions and mathematical statement

Let $C$ and $D$ be irreducible components of Lusztig nilpotent varieties of Dynkin type $A$, in the common representation-theoretic setting intended by the supplied source. Write $\pi(C)$ and $\pi(D)$ for their associated simple modules. For a module $M$, its socle $\operatorname{soc}(M)$ is the sum of its simple submodules.

The supplied conjecture asserts that, for every such pair $C,D$, \[ C\text{ is rigid}\quad\Longrightarrow\quad \operatorname{soc}\left(\pi(C)\times\pi(D)\right) \cong \pi(C*D). \] Here the displayed isomorphism expresses the source's equality of representations up to isomorphism. In particular, the assertion says that this socle is simple and has the isomorphism class attached to $C*D$. Rigidity is required only of $C$; the supplied statement imposes no rigidity hypothesis on $D$.

Unresolved formulation qualifications. The record does not define rigidity, the assignment $C\mapsto\pi(C)$, the ordered component operation $*$, or the representation product $\times$. It also does not specify the underlying field, ambient module category, dimension-vector conventions, or compatibility conditions between the varieties containing $C$ and $D$. These symbols retain their intended source-specific meanings; their definitions and conventions must be recovered from the cited sources before this becomes a fully specified standalone conjecture. The order of both products is retained exactly as supplied.

Short English statement

The conjecture says that if the first Lusztig component $C$ is rigid, then the product of the representations attached to $C$ and $D$ has a simple socle, namely the representation attached to the ordered component product $C*D$. The precise representation-theoretic constructions are missing from the supplied record and require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006598$, title Socles of Products from Lusztig Components, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record lists proposed year 2026 and supplies no named proposer.
• Representation Theory of Quivers and Finite-Dimensional Algebras (2026), Oberwolfach Reports, Report 6/2026, pp. 345--432, DOI 10.4171/owr/2026/6. Supplied report and literature-source link: https://doi.org/10.4171/owr/2026/6. The extracted passage says that the conjecture was made in reference [LM25]; the record does not supply a complete bibliographic entry under that label.
• Supplied literature source: Lapid--Minguez, binary operation on nilpotent varieties II, https://arxiv.org/abs/2111.05162.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, says that the socle identity is still called a conjecture in the 2026 report, and says that the primary paper verifies only special cases, including a stated rigidity class. This is supplied dated curation, not an independently verified literature assessment; the special cases and rigidity class are not specified in the record.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is supplied verification metadata, not an independent verification of this draft or of mathematical correctness.