Nonnegativity in Complete Indices of Bruhat Intervals

Definitions and mathematical statement

Let $(W,S)$ be the Coxeter system underlying the source's Bruhat intervals, let $\ell$ be its length function, and let $\leq$ be Bruhat order. For $u\leq v$, the interval is $[u,v]=\{x\in W:u\leq x\leq v\}$. For $u<v$, put $r=\ell(v)-\ell(u)$.

The complete $cd$-index is a polynomial in noncommuting variables $c,d$, with weighted degrees $\deg c=1$ and $\deg d=2$. Use the source's convention for this complete index, denoted here by $\Psi^{\mathrm{comp}}_{u,v}$. The usual interval $cd$-index is its top homogeneous part of weighted degree $r-1$. The source's lower-degree part is therefore \[ \widetilde\Psi_{u,v} =\sum_{\deg w<r-1}a_w\,w\in{\mathbb{Z}}\langle c,d\rangle, \] where $w$ ranges over words in $c,d$ and $a_w$ is its coefficient in the complete index.

The question asks whether, for every intended Bruhat interval, \[ [w]\widetilde\Psi_{u,v}=a_w\geq0 \qquad\text{for every word }w\text{ of weighted degree below }r-1. \] This is coefficientwise nonnegativity in the noncommutative $cd$-basis. It is not merely nonnegativity of a specialization or of the top ordinary $cd$-index. The extract does not specify a Coxeter-group subclass or reproduce the complete-index construction and normalization; the standard weighted grading above is explicit. Degenerate intervals require the source's unit convention and do not carry a positive-degree lower-part condition under the usual empty-lower-part convention.

Short English statement

Are all lower-weighted-degree coefficients of the complete $cd$-index of every Bruhat interval nonnegative? The question concerns each noncommutative coefficient, beyond the top ordinary index.

Sources

• Supplied report: Geometric and Topological Combinatorics (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/04, pp. 195--272; proposed year 2007. https://doi.org/10.4171/owr/2007/04.
• Local UnsolvedMath registry, permanent ID 30000682, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the complete coefficient question was preserved while notation was restored. The supplied 21 August 2026 literature triage labels the extracted claim open and reports no located authoritative complete resolution. No additional external reference is supplied; the dated assessment is not independently updated here.