For every integer n≥ 1, every integer 1≤ r≤ n, every real matrix A=(a_ij)∈R^n× n whose minors of orders at most r are nonnegative, and every partition lambda vdash n with ell(lambda)≤ r, is the monomial immanant phi^lambda[A]≥ 0, where phi^lambda is the class function dual to the Young permutation character and phi^lambda[A]=∑_sigma∈mathfrakS_nphi^lambda(sigma)∏_i=1^n a_i sigma(i)?

Stembridge’s monomial-immanant nonnegativity problem

For every integer $n\ge 1$ , every integer $1\le r\le n$ , every real matrix $A=(a_{ij})\in\mathbb{R}^{n\times n}$ whose minors of orders at most $r$ are nonnegative, and every partition $\lambda\vdash n$ with $\ell(\lambda)\le r$ , is the monomial immanant $\phi^\lambda[A]\ge 0$ , where $\phi^\lambda$ is the class function dual to the Young permutation character and $\phi^\lambda[A]=\sum_{\sigma\in\mathfrak{S}_n}\phi^\lambda(\sigma)\prod_{i=1}^n a_{i\sigma(i)}$ ?

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Schur-positivity formulation
For every skew partition $\lambda/\mu$ and every partition $\nu$ , is the immanant $\operatorname{Imm}_{\phi^\nu}(H_{\lambda/\mu})$ Schur-positive?
source: Background supplied with the source

References

Primary source

arXiv

Additional references

On the nonnegativity of monomial immanants for hook partitions — arXiv — Xiangshuai Dong, Tingzeng Wu, Xing Gao

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new paper proves the conjecture for one important family of shapes, but the full problem for arbitrary shapes remains open.

Stembridge’s problem asks whether $\operatorname{Imm}_{\phi^\nu}(H_{\lambda/\mu})$ is Schur-positive for every skew partition $\lambda/\mu$ and partition $\nu$ . The general assertion remains untreated beyond substantial special cases.

Known results

Haiman proved Schur-positivity for immanants associated with arbitrary symmetric-group characters.
Dong, Wu, and Gao (2023) proved the hook-partition case using nonnegative combinations of Stanley–Stembridge characters.

September 2026 hook-partition result

A September 2026 preprint by Xiangshuai Dong, Tingzeng Wu, and Xing Gao reports nonnegativity for all hook partitions. This confirms progress on the structured hook family, but does not address arbitrary partitions; the reported advance is unverified here.

Current status (as of September 2026): The hook-partition case is claimed in specialist preprints, while arbitrary partitions remain open and no general resolution is recorded.

Sources

No solutions have been posted yet.

Schur-positivity formulation

Primary source

Additional references

Progress summary