Representation Models for Toric Cohomology and Symmetric Functions

Definitions and mathematical statement

Let $B_n$ be the Boolean lattice of subsets of $[n]$ and let $C_n$ be an $n$-element chain. The product symbol in the source is interpreted as the Rees product of ranked posets: $P*Q$ consists of pairs $(p,q)$ with $\rho_P(p)\geq\rho_Q(q)$, ordered by \[ (p,q)\leq(p',q')\iff p\leq p',\ q\leq q',\ \rho_P(p')-\rho_P(p)\geq\rho_Q(q')-\rho_Q(q). \] Put $P_n=(B_n\setminus\{\widehat0\})*C_n$. The symmetric group $S_n$ acts on subsets and fixes the chain coordinate. With ranks $\rho(A)=|A|-1$ and chain coordinates $j=0,\ldots,n-1$, its maximal elements are $m_j=([n],j)$. After adjoining a minimum, the corresponding maximal open intervals are $I_{n,j}=(\widehat0,m_j)$; their order complexes carry induced $S_n$-representations on reduced homology.

Let $X_n$ be the smooth projective toric variety of the type-$A_{n-1}$ Coxeter fan, with its natural $S_n$-action. The first task is a direct, explicitly constructed equivariant isomorphism between its cohomology representation and the sign twist of the interval-homology representation whose Frobenius characteristics the report identifies. In standard indexing the component comparison takes the form \[ H^{2j}(X_n;{\mathbb{C}})\cong \operatorname{sgn}_{S_n}\otimes \widetilde H_{n-2}(\Delta(I_{n,j});{\mathbb{C}}), \qquad0\leq j\leq n-1. \] The component indexing and small-$n$ conventions must agree with the report; they are not specified in the extract. An abstract existence deduction from equality of characters is not the requested direct construction.

For a partition $\lambda\vdash n$ and an admissible index $j$, the second task asks for an actual finite-dimensional $S_n$-representation $V_{\lambda,j}$ with \[ \operatorname{ch}(V_{\lambda,j})=Q_{\lambda,j}. \] For a representation $V$, its Frobenius characteristic is $\operatorname{ch}(V)=\sum_{\mu\vdash n}\chi_V(\mu)p_\mu/z_\mu$, where $p_\mu$ is the power-sum symmetric function and $z_\mu$ the centralizer order of a permutation of cycle type $\mu$. Thus the existence question is equivalent to Schur expansion coefficients of $Q_{\lambda,j}$ being nonnegative integers. The source's partition-indexed Eulerian quasisymmetric function is retained; the extract gives no defining formula, index range or normalization for it. It is not replaced by the unrefined Eulerian function or by a virtual character.

Short English statement

Construct the direct equivariant correspondence between type-$A$ toric cohomology and sign-twisted Rees-product interval homology. Separately, realize every intended partition-indexed $Q_{\lambda,j}$ as the Frobenius characteristic of an actual symmetric-group representation.

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.
• Supplied reference labelled "Eulerian quasisymmetric functions and cyclic sieving": https://math.miami.edu/~wachs/papers/cyclic.pdf.
• Local UnsolvedMath registry, permanent ID 30000692, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the official report confirms both questions. The original says the two representations' Frobenius characteristics are identified there. The supplied 21 August 2026 literature triage labels the extraction open and reports refinements and special cases without a general representation realization of $Q_{\lambda,j}$. The assessment emphasizes the second question and does not independently settle the first. No PDF access or updated literature assessment is asserted here.