Geometric Interpretations of Coxeter-Group R-Polynomials

Definitions and mathematical statement

Let $(W,S)$ be a Coxeter system, with length $\ell$ and Bruhat order $\leq$. In the ordinary one-parameter normalization, the $R$-polynomials $R_{u,v}(q)\in{\mathbb{Z}}[q]$ satisfy $R_{u,u}=1$, $R_{u,v}=0$ for $u\not\leq v$, and, for $\ell(vs)<\ell(v)$, \[ R_{u,v}(q)= \begin{cases} R_{us,vs}(q),&\ell(us)<\ell(u),\\ qR_{us,vs}(q)+(q-1)R_{u,vs}(q),&\ell(us)>\ell(u). \end{cases} \] This fixes a standard meaning of the symbols, including $R_{e,s}=q-1$. The source does not specify this normalization, variants such as transformed $\widetilde R$-polynomials, or the class of Coxeter systems; those choices must be checked against its missing context.

The request is for a geometric interpretation or geometric connections for these polynomials. It does not prescribe a variety, invariant, functor, point-counting formula, cohomological realization or compatibility to be proved. A finite crystallographic Weyl-group setting with flag varieties, a Kac--Moody setting and arbitrary Coxeter groups are different scopes. Nor may an interpretation for Kazhdan--Lusztig polynomials simply be substituted for one for $R$-polynomials. If a cohomological interpretation is intended, ordinary Betti numbers, alternating/compact-support data and point counts must be distinguished, together with any parameter substitutions.

A precise theorem would therefore require a specified geometric object assigned to $(u,v)$ and an equality or structural comparison with $R_{u,v}$, for a stated class of groups and normalization. No such assignment or comparison is supplied, and none is invented here. The question is preserved as a research program rather than a single quantified existence theorem with a unique target.

The literature triage checked 21 August 2026 explicitly says the fragment lacks surrounding definitions and is too vague to match to a unique theorem or conjecture. The open label is not a claim that no geometric interpretation is known in any setting. The corrected-verified wording is reported provenance, not evidence that the intended geometric target has been recovered.

Short English statement

The draft fixes one standard $R$-polynomial convention, but the geometric target and Coxeter-group range remain unspecified. The source itself says the fragment cannot be matched to one precise conjecture. Its open label does not deny known interpretations in particular settings.

Sources

• Local UnsolvedMath registry, permanent ID 30000076, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• Supplied citation: Combinatorial Commutative Algebra (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/32, pp. 1703--1746: https://doi.org/10.4171/owr/2004/32. Proposed year is separately 2004.
• Supplied report reference: https://oa.tib.eu/renate/server/api/core/bitstreams/796d4bc0-b3ab-46f9-b550-5b437dd06ff1/content.
• Literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. It supplies no specific geometry theorem or additional authoritative bibliography.