Colored Realizations of Gorenstein* $cd$-Indices

Definitions and mathematical statement

For a finite bounded graded poset $P$ with minimum $\hat 0$, maximum $\hat 1$, and $\operatorname{rank}(\hat 1)=n+1$, let $K(P)$ be the order complex of $P\setminus\{\hat 0,\hat 1\}$: its faces are chains in this proper part. Under the usual interpretation over a field $k$, the Gorenstein* condition means that $K(P)$ is a homology sphere over $k$, including the corresponding sphere-homology condition for the link of every face. The supplied record does not specify the coefficient field or whether the condition is intended over every field; this qualification remains unresolved.

For $T\subseteq[n]=\{1,\ldots,n\}$, let $f_T(P)$ count chains in the proper part whose set of ranks is exactly $T$, with $f_{\varnothing}(P)=1$, and put \[ h_T(P)=\sum_{U\subseteq T}(-1)^{|T|-|U|}f_U(P). \] The $ab$-index is \[ \Psi_P(a,b)=\sum_{T\subseteq[n]}h_T(P)u_T, \qquad u_T=u_1\cdots u_n,\quad u_i=\begin{cases}b,&i\in T,\\a,&i\notin T.\end{cases} \] Here $a,b$ are noncommuting variables. The $cd$-index $\Phi_P(c,d)$ is the polynomial characterized by \[ \Psi_P(a,b)=\Phi_P(a+b,ab+ba). \] Write \[ \Phi_P(c,d)=\sum_{\deg w=n}\alpha_w(P)w, \qquad \deg c=1,\quad\deg d=2, \] where $w$ ranges over noncommutative $cd$-monomials of weighted degree $n$.

For a finite simplicial complex $\Delta$ and a vertex coloring $\kappa:V(\Delta)\to[m]$, define \[ f_S(\Delta,\kappa)=\#\{F\in\Delta:\{\kappa(v):v\in F\}=S\}, \qquad S\subseteq[m]. \] The empty face is included, so $f_{\varnothing}(\Delta,\kappa)=1$. A coloring is proper if its restriction to each face is injective. The record requires $m=\dim(\Delta)+1$, but does not explicitly state properness.

Intended realization question, with unresolved specifications. For every Gorenstein* poset $P$, can its entire $cd$-index be realized by the flag face numbers of a colored simplicial complex $\Delta$ with $m=\dim(\Delta)+1$ colors? To express this as coefficient equalities would require a prescribed correspondence $w\mapsto\iota(w)\subseteq[m]$, giving \[ \alpha_w(P)=f_{\iota(w)}(\Delta,\kappa) \quad\text{for every }cd\text{-monomial }w\text{ of weighted degree }n. \] This display is schematic: the supplied record does not define $\iota$, relate $m$ to $n$, or state conditions on flag entries outside this correspondence. The correspondence is not being introduced as an arbitrary existential choice. Together with the unspecified Gorenstein* coefficient convention and coloring convention, these omissions prevent a unique precise realization conjecture from being recovered from the supplied text.

Short English statement

The supplied question asks whether every Gorenstein* poset has its full $cd$-index represented by flag face counts of a simplicial complex colored with $\dim(\Delta)+1$ colors. The coefficient-to-color-subset correspondence, the relation between poset rank and the number of colors, and certain underlying conventions are missing, so the target remains an incomplete formulation requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001701$, source problem code catalogue entry, title: Colored Realizations of Gorenstein* $cd$-Indices. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2011$; no individual proposer is supplied.
• Supplied source citation: Topological and Geometric Combinatorics (2011), Oberwolfach Reports, pp. 349--423, DOI 10.4171/owr/2011/08. Source URL and supplied literature-source link, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/08. No item-specific page or named contributor is provided.
• The supplied literature assessment was checked on 2026-08-21. It labels the problem open, describes the extracted item as an incomplete statement of the general $cd$-index realization conjecture, and reports that no general resolution was identified. This is supplied dated curation, not an independently verified assessment.
• The registry labels the statement corrected_verified and says it was checked against the official report. These are supplied provenance claims; no independent source verification has been performed for this draft.