Categorical Diagonalization of Coxeter Full Twists

Definitions and mathematical statement

Let $(W,S)$ be a finite Coxeter system, and let $B_W$ be its Artin braid group. Write $w_0$ for the longest element of $W$, and $\boldsymbol{w}_0\in B_W$ for its positive braid lift. Under the usual positive-full-twist convention, the full twist is \[ T=\boldsymbol{w}_0^{\,2}. \] Write $F_T=F(T)$ for the Rouquier complex associated with this braid in the intended graded monoidal homotopy category. The supplied record does not specify that category, the realization of $W$, or the coefficient ring.

For a specified Hecke algebra of $(W,S)$ with its Kazhdan--Lusztig basis, the two-sided cells are the equivalence classes of the two-sided preorder generated by occurrence of basis elements under left and right multiplication. Denote their set by $\mathfrak C(W)$. The Hecke parameters needed to determine this cell structure are not explicitly supplied.

Source conjecture. For every finite Coxeter group $W$, is $F_T$ categorically diagonalizable, with its categorical spectrum in bijection with $\mathfrak C(W)$, and with the eigenshift assigned to each cell equal to the shift predicted for that cell?

Here an eigenshift refers to a scalar object obtained by shifting the monoidal unit in the relevant gradings. If internal and homological shifts are denoted by $(a)$ and $[b]$, respectively, such an object is written $\mathbf 1(a)[b]$. This notation does not specify the intended shift values. The record calls these eigenshifts both "known" and "predicted", but supplies neither their formulas nor their normalization.

Unresolved formulation. The source does not define categorical diagonalizability, its categorical spectrum, or the ambient category or completion in which any required projectors exist. Consequently, the question above preserves the supplied universal claim, but is a schematic conjecture rather than a fully specified mathematical proposition. Its precise interpretation requires the intended categorical definition, coefficient and realization hypotheses, Hecke parameters, and cellwise eigenshift formulas; these cannot be reconstructed uniquely from this record.

Short English statement

The supplied conjecture asks whether every finite Coxeter group's full-twist Rouquier complex admits a categorical diagonalization indexed by two-sided cells, with prescribed grading shifts. The categorical setting and shift formulas are missing, so the formulation requires mathematical review.

Sources

• Supplied local problem-registry record, permanent UnsolvedMath ID $30002882$, source code catalogue entry, title: Categorical Diagonalization of Coxeter Full Twists; collection: Oberwolfach Reports Open Problems.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2016), Oberwolfach Reports, pp. 1385--1447, DOI 10.4171/owr/2015/25. https://doi.org/10.4171/owr/2015/25. The record assigns proposed year 2015 and names no individual proposer. The cited report documents workshop discussions; no problem-specific page or named contributor is supplied.
• Elias--Hogancamp, On the full twist, supplied literature reference: https://arxiv.org/abs/1801.00191.
• Elias--Hogancamp, "published full-twist work", supplied reference label: https://doi.org/10.1215/00127094-2024-0078.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports categorical diagonalization in type A and several small or dihedral cases, while describing the assertion for every finite Coxeter group as open. This is supplied dated curation, not an independently verified assessment of the literature or the cited results.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of the cleanup. This is supplied registry metadata, not independent mathematical verification of this draft.