L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910178660581376 |
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| author | Speyer, David E |
| author_facet | Speyer, David E |
| contents | Let $λ$, $μ$, $λ'$, $μ'$ be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if $λ+μ= λ' + μ'$, and $\min(λ_i-λ_j, μ_i-μ_j) \leq λ'_i - λ'_j \leq \max(λ_i-λ_j, μ_i-μ_j)$ for all $1 \leq i<j \leq n$, then $s_{λ'} s_{μ'} - s_λ s_μ$ is Schur nonnegative. We prove this conjecture.
Our proof is based on two key ideas. First, we introduce a new combinatorial model for Littlewood-Richardson coefficients which we name ``skeps", which are similar to but distinct from Knutson and Tao's hives. Second, we use tools from Murota's theory of L-convexity to prove an L-log-concavity theorem for skeps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05007 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy Speyer, David E Combinatorics Let $λ$, $μ$, $λ'$, $μ'$ be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if $λ+μ= λ' + μ'$, and $\min(λ_i-λ_j, μ_i-μ_j) \leq λ'_i - λ'_j \leq \max(λ_i-λ_j, μ_i-μ_j)$ for all $1 \leq i<j \leq n$, then $s_{λ'} s_{μ'} - s_λ s_μ$ is Schur nonnegative. We prove this conjecture. Our proof is based on two key ideas. First, we introduce a new combinatorial model for Littlewood-Richardson coefficients which we name ``skeps", which are similar to but distinct from Knutson and Tao's hives. Second, we use tools from Murota's theory of L-convexity to prove an L-log-concavity theorem for skeps. |
| title | L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.05007 |