MathDB

Lam–Postnikov–Pylyavskyy conjecture on Schur positivity

Let λ\lambda, μ\mu, λ′\lambda', and μ′\mu' be partitions in Zn\mathbb{Z}^n. For vectors x,y∈Znx,y\in\mathbb{Z}^n, define

Π(x,y)={z∈Zn:min⁡(xi−xj,yi−yj)≤zi−zj≤max⁡(xi−xj,yi−yj) for all 1≤i<j≤n}.\Pi(x,y)=\{z\in\mathbb{Z}^n:\min(x_i-x_j,y_i-y_j)\le z_i-z_j\le\max(x_i-x_j,y_i-y_j)\text{ for all }1\le i<j\le n\}.

A symmetric polynomial is Schur nonnegative if it is a nonnegative linear combination of Schur polynomials. Lam–Postnikov–Pylyavskyy's conjecture. If λ′+μ′=λ+μ\lambda'+\mu'=\lambda+\mu and λ′∈Π(λ,μ)\lambda'\in\Pi(\lambda,\mu), then

sλ′sμ′−sλsμs_{\lambda'}s_{\mu'}-s_{\lambda}s_{\mu}

is Schur nonnegative. The paper presents a proof using skeps and LL-log-concavity, so the conjecture is resolved.

References

Primary source

David E Speyer, “L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy”, arXiv:2601.05007 (2026).

Additional references

2 papers in this index state this conjecture (2025–2026). The statement above is taken from the most recent of them; the others are arXiv:2506.00349.

log-concavityschur-polynomialssymmetric-functions
Posted

Progress summary

Refreshed
Claimed solved

A paper by David Speyer gives a proof of the conjecture, and a later paper treats it as resolved while extending it.

The conjecture predicts that, when two pairs of partitions have the same total and one pair lies between the other in the prescribed sense, the corresponding difference of Schur-function products has only nonnegative Schur coefficients. It was formally stated by Dobrovolska and Pylyavskyy after earlier work by Lam, Postnikov, and Pylyavskyy.

August 2026 resolution

David Speyer’s preprint claims the full coefficientwise inequality cλμν≤cλ′μ′νc_{\lambda\mu}^{\nu}\le c_{\lambda'\mu'}^{\nu}, using skeps and LL-log-concavity. A subsequent arXiv paper states that Speyer recently resolved the conjecture and develops a skew-Schur generalization, providing independent corroboration.

Current status (as of August 2026): The ordinary Lam–Postnikov–Pylyavskyy conjecture is resolved by Speyer’s claimed proof and its subsequent treatment as settled; the skew generalization is also asserted in the later paper.

Sources

Solutions 0

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