For every integer n≥ 1 and every pair of partitions mu⊆lambda, let s_lambda/mu(x_1,…,x_n)=∑_alpha∈Z_≥ 0^nc_alpha x^alpha be the skew Schur polynomial, where x^alpha=x_1^alpha_1… x_n^alpha_n. Define its normalization by N(s_lambda/mu)=∑_alpha fracc_alphaalpha!x^alpha, where alpha!=alpha_1!…alpha_n!. The conjecture asserts that N(s_lambda/mu) is a Lorentzian polynomial.

The Lorentzian conjecture for normalized skew Schur polynomials

For every integer $n\ge 1$ and every pair of partitions $\mu\subseteq\lambda$ , let $s_{\lambda/\mu}(x_1,\ldots,x_n)=\sum_{\alpha\in\mathbb{Z}_{\ge 0}^n}c_\alpha x^\alpha$ be the skew Schur polynomial, where $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ . Define its normalization by $\mathcal{N}(s_{\lambda/\mu})=\sum_{\alpha} \frac{c_\alpha}{\alpha!}x^\alpha$ , where $\alpha!=\alpha_1!\cdots\alpha_n!$ . The conjecture asserts that $\mathcal{N}(s_{\lambda/\mu})$ is a Lorentzian polynomial.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Normalized skew Schur polynomials are Lorentzian — arXiv

algebraic-combinatorics lorentzian-polynomials symmetric-functions

1 week ago

Progress summary

Refreshed 5 days ago

Claimed solved

Two August 2026 preprints claim to prove the conjecture, but no independent verification has yet been reported.

The conjecture, posed by Huh, Matherne, Mészáros, and St. Dizier, asserts that normalized skew Schur polynomials are Lorentzian. It would imply new log-concavity inequalities for skew Kostka numbers.

Known results

The ordinary case was proved, with strong log-concavity for normalized Schur polynomials and corresponding Kostka-number inequalities (Huh et al., 2019).
The general skew statement had computational support but was conjectural in the 2019 paper.

August 2026 claimed proofs

A preprint dated 12 August claims the full result by specializing Schubert polynomials and applying duality. A second August preprint independently claims the skew-Schur result through Richardson volume polynomials and strengthens it to skew Schur $P/Q$ -functions; both remain unrefereed and unverified.

Current status (as of August 2026): The conjecture is claimed solved by two unrefereed arXiv preprints, but independent mathematical verification is absent, so the proof remains unconfirmed.

Sources

No solutions have been posted yet.