Problem 6.5 — (S.J.

Definitions and mathematical statement

The braid group $B_n$ has generators $\sigma_1,\ldots,\sigma_{n-1}$, with relations \[ \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1},\qquad \sigma_i\sigma_j=\sigma_j\sigma_i\quad(|i-j|\geq2). \] A representation $\rho:B_n\to\operatorname{GL}(V)$ is irreducible if $V$ has no nonzero proper invariant subspace. The Birman--Murakami--Wenzl (BMW) algebra is a parameter-dependent quotient of the braid group algebra; its modules give braid-group representations through the quotient map.

S. J. Bigelow asks for a larger family of irreducible $B_n$-representations containing those arising from BMW algebras. The natural construction target is an explicitly described family with the BMW representations recovered up to isomorphism in the specified parameter regime and with genuinely additional irreducible representations. Merely naming all irreducible representations as a family would not supply the requested construction.

The source does not fix a coefficient field, finite-dimensionality, BMW parameters or generic versus specialized algebras. It also does not specify whether inclusion means actual members or specializations of a parameterized family, nor a measure of how much larger the family should be. No faithfulness, unitarity, classification of every irreducible, or compatibility between different strand numbers is added as a requirement.

Short English statement

Construct an explicit family of irreducible braid-group representations that extends the BMW-derived family. The field, parameter regime, dimension class and meaning of inclusion need source conventions.

Sources

• Local UnsolvedMath registry, permanent ID 10400113, source code catalogue entry; statement attribution S. J. Bigelow; proposer field and proposal year absent; update 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 6.5, PDF page 97: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Present primary-literature status and redistribution terms require review.
• The supplied summary repeats the extension-of-family question without exact additional references or a verified resolution.