For every n≥ 1 and every partition lambda vdash n, let m_lambda be the monomial symmetric function and let operatornameImm_m_lambda denote the corresponding monomial immanant. Then there exist integers c_lambda,w≥ 0, indexed by w∈ S_n, such that operatornameImm_m_lambda(X)=∑_w∈ S_nc_lambda,woperatornameImm_w(X), where operatornameImm_w is the Kazhdan–Lusztig immanant associated with w. Equivalently, every monomial immanant is a non-negative integer combination of Kazhdan–Lusztig immanants at q=1.

Haiman–Stembridge monomial-immanant positivity conjecture

For every $n\ge 1$ and every partition $\lambda\vdash n$ , let $m_\lambda$ be the monomial symmetric function and let $\operatorname{Imm}_{m_\lambda}$ denote the corresponding monomial immanant. Then there exist integers $c_{\lambda,w}\ge 0$ , indexed by $w\in S_n$ , such that $\operatorname{Imm}_{m_\lambda}(X)=\sum_{w\in S_n}c_{\lambda,w}\operatorname{Imm}_w(X)$ , where $\operatorname{Imm}_w$ is the Kazhdan–Lusztig immanant associated with $w$ . Equivalently, every monomial immanant is a non-negative integer combination of Kazhdan–Lusztig immanants at $q=1$ .

References

Hook monomial immanants are Kazhdan-Lusztig positive — arXiv — Keith McNulty

Source: arXiv

algebraic-combinatorics representation-theory

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Open · progress reported

A new result proves the conjecture for every hook-shaped case, but the general case covering arbitrary partitions remains open.

Haiman’s 1993 conjecture asserts positivity and unimodality for coefficients of certain Kazhdan–Lusztig monomial-character polynomials. It is stronger than the related totally nonnegative-matrix conjecture and remains unresolved for arbitrary partitions.

Known results

Clearman, Hyatt, Shelton, and Skandera showed that the conjectured positivity would imply the Shareshian–Wachs and Stanley–Stembridge conjectures.
Stembridge established several special cases, including rectangular partitions, the partition $(2,1^{n-2})$ , and small cases.
A 2023 result proved a different hook-shaped Schur-positivity statement for Jacobi–Trudi matrices.

October 2026 hook-family result

Keith McNulty’s reported result establishes Kazhdan–Lusztig positivity for all hook shapes, with related positivity for associated functions and products. A separate September 29, 2026 paper proves the corresponding bounded-minor and totally nonnegative cases for every hook partition. These are substantial partial results, not a proof for arbitrary partitions, and the claims remain unverified here.

Current status (as of October 2026): positivity is established or claimed for important hook families, while the Haiman–Stembridge conjecture for arbitrary partitions remains open.

Sources

No solutions have been posted yet.