Question 5 — Is there a homological definition of representations of the Birman-Wenzl- Murakami algebra?

Definitions and mathematical statement

Question 5. Is there a homological definition of representations of the Birman--Wenzl--Murakami algebra? The source expresses belief that the answer is yes, but does not supply the intended homology spaces, coefficient/local-system data, maps, or factorization through the algebra. A precise construction must specify those data and verify the defining relations; no homological model is asserted here.

The accompanying source proposes an algebra $Z_n$ generalizing Iwahori--Hecke and Birman--Wenzl--Murakami algebras. Retain $R,q,t$ as its unspecified coefficients and parameters. It abbreviates $\sigma_{i_1\cdots i_k}=\sigma_{i_1}\cdots\sigma_{i_k}$ and uses barred words for inverse notation. The extraction does not clearly distinguish inverse of the entire word from the product of inverses in the written order; bars below retain the source notation pending that clarification.

In the stated braid-group algebra $R B_n$, it defines \[ X_2=q\bar\sigma_1+1-q-\sigma_1, \qquad X_k=\left(q^{k-1}\bar\sigma_{(k-1)\cdots1} -\sigma_{1\cdots(k-1)}\right)X_{k-1}\quad(3\leq k\leq n). \] Thus the displayed initial cases are \[ X_3=(q^2\bar\sigma_{21}-\sigma_{12})X_2, \qquad X_4=(q^3\bar\sigma_{321}-\sigma_{123})X_3. \] The proposed $Z_n$ is the quotient by the displayed relations \[ (q\bar\sigma_2+1-q-\sigma_2)X_2 =(q\bar\sigma_{21}-\sigma_{12})X_2, \] \[ (q^2\bar\sigma_{32}-\sigma_{23})X_3 =(q^2\bar\sigma_{321}-\sigma_{123})X_3, \] \[ (q^3\bar\sigma_{432}-\sigma_{234})X_4 =(q^3\bar\sigma_{4321}-\sigma_{1234})X_4, \] and the continuation terminating, as printed, with \[ (q^{n-1}\bar\sigma_{n\cdots2}-\sigma_{2\cdots n})X_n =(q^{n-1}\bar\sigma_{n\cdots1}-\sigma_{1\cdots n})X_n. \] There is an indexing conflict: standard $B_n$ has generators only $\sigma_1,\ldots,\sigma_{n-1}$, while the final relation uses $\sigma_n$. The braid-index or relation-range convention must be repaired from the source before this is a complete algebra presentation; no silent replacement by $B_{n+1}$ or truncation is made.

The source states that $H_n(1,-q)$ is obtained by imposing $X_2=0$, and the Birman--Wenzl--Murakami algebra by imposing $X_3=0$ and $\sigma_1X_2=tX_2$. These quotient identifications retain the unresolved parameter and index conventions. It calls the homological motivation unclear and unpublished at the time, suggests combinatorial computations, and introduces a following question about whether $Z_n$ is bigger than the Birman--Wenzl--Murakami algebra; that following precise question is not supplied.

Short English statement

Seek a homological construction of Birman--Wenzl--Murakami representations. The source's proposed algebra $Z_n$ and quotient formulas are retained as context, with unresolved inverse-word notation, coefficients, and a final braid-generator indexing conflict.

Sources

• Local UnsolvedMath registry, permanent ID 11000262, source code catalogue entry, Question 5, PDF page 322. The chapter header identifies S. Bigelow; no individual proposer field is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across chapters, with repeated labels disambiguated by page.
• The source supplies the $X_k$ recursion, relations, and claimed Hecke/Birman--Wenzl--Murakami quotients directly, without a separate citation. It explicitly describes its homological motivation as then unclear and unpublished.
• Supplied status open and OPEN-TRIAGE. The summary says only "Cannot verify." Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.