Extension of Representations along $\tau$

Definitions and mathematical statement

Let $K,N,M$, a map $\tau:K\to N$, and a representation $\sigma:K\to M$ be the objects intended in the source. The extension problem asks for conditions equivalent to \[ \exists\rho:N\to M\text{ an admissible global representation} \quad\text{such that}\quad \rho(\tau(x))=\sigma(x)\quad\text{for every }x\in K. \] Thus "$\sigma$ is extensible to $N$ via $\tau$" means that this factorization exists. If the relevant representation classes admit composition, define \[ \operatorname{res}_{\tau}:\operatorname{Rep}(N,M)\longrightarrow\operatorname{Rep}(K,M), \qquad\rho\longmapsto\rho\circ\tau. \] The question is to characterize when $\sigma$ lies in its image. This notation does not assume that representation means a group homomorphism into a matrix group; in a lattice workshop it could instead mean representation of forms or lattices by structure-preserving maps. The underlying category and the meaning of representation are not specified in the record.

The original passage additionally defines extension at a prime $p$, referring to maps $\rho^{(p)}:N_p\to M_p$ and a missing commutative diagram. If source-defined localizations and induced maps $\tau_p,\sigma_p$ are available, the local condition would be $\rho^{(p)}\circ\tau_p=\sigma_p$. The record does not define $N_p,M_p,K_p$, identify the relevant primes, or specify whether completions or localizations are intended. It does not assert that local extensions at all primes imply a global extension; such a local-to-global principle would be an additional theorem to formulate and test, not a consequence of the definition.

The supplied literature triage checked 2026-08-21 says that the ambient objects, assumptions, and target theorem are too underspecified to identify a unique literature problem. Accordingly, the factorization question and its local terminology are preserved, without inventing hypotheses that make it a particular representation-extension conjecture.

Short English statement

Characterize when the given representation factors through $\tau$ as $\sigma=\rho\circ\tau$ for a global representation on $N$. The structures being represented and the local-at-$p$ conventions are absent, so no definite local-to-global theorem can yet be stated.

Sources

• Local UnsolvedMath registry, permanent ID 30000168, catalogue entry, Extension of Representations along $\tau$.
• Gitter und Anwendungen (2005), Oberwolfach Reports, pp. 5--67, DOI https://doi.org/10.4171/owr/2005/01.
• The supplied original passage includes the global and at-$p$ extension definitions and refers to an absent Diagram (*). No separate authoritative theorem citation is supplied.
• Supplied literature triage checked 2026-08-21 identifies the incomplete formulation; the corrected statement retains the explicit factorization question.