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L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy

David E Speyer

Abstract

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.

L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy

Abstract

Let , , , be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if , and for all , then 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.
Paper Structure (7 sections, 26 theorems, 71 equations, 6 figures)

This paper contains 7 sections, 26 theorems, 71 equations, 6 figures.

Table of Contents

  1. $L$-convexity and $L \textrm{-} \! \log$-concavity
  2. Overview of the proof
  3. Hives, the octahedron recurrence, and proof of Theorem \ref{['SkepLR']}
  4. More on $L$-convexity
  5. Marginalizations of $L \textrm{-} \! \log$-concave functions are $L \textrm{-} \! \log$-concave
  6. Proofs of Theorems \ref{['SkepExtLConcave']}, \ref{['BetterLPP']} and the LPP conjecture
  7. The inequalities necessary to prove the LPP conjecture

Key Result

Theorem \ref{SkepLR}

Let $\lambda$, $\mu$ and $\nu$ be partitions. Then $c_{\lambda \mu}^{\nu}$ is the number of skeps with $g_{n0}=0$, $\partial^{\hbox{$\nwarrow$}}(g) = (\nu_1, \nu_2, \ldots, \nu_n)$ and $\partial^{}(g) = (\lambda_1, \mu_1, \lambda_2, \mu_2, \ldots, \lambda_n, \mu_n)$.

Figures (6)

  • Figure 2.1: The left hand side depicts the skep inequalities: In each green parallelogram, and in all translates thereof, the sum of the $+$ vertices is more than the sum of the $-$ vertices; we also impose this condition on the green line segment. The right hand side, with the pink parallelograms, depicts the hive inequalities in the same manner.
  • Figure 3.1: The restriction of the octahedron recurrence to the walls of $T_4$. The four corners of the figure are labeled by the vertices of the tetrahedron $T_4$. Fold along the dashed line; map the left side linearly to the wall $i=0$ and map the right side linearly to the wall $j=0$. Each edge is labelled with $\widetilde{h}_{\text{left endpt}} - \widetilde{h}_{\text{right endpt}}$. The red edges indicate $S_{\text{hive}}^{\text{bottom}}$, $S_{\text{skep}}^{\text{bottom}}$, $S_{\text{skep}}^{\text{top}}$ and $S_{\text{hive}}^{\text{top}}$.
  • Figure 3.2: The projections of $\mathcal{S}^{\text{top}}$, $\mathcal{S}_1$, $\mathcal{S}_2$ and $\mathcal{S}'_1$ to the $(i,j)$-plane (for $n=4$). The numbers indicate the value of the $t$-coordinate.
  • Figure 3.3: The intersections $\mathcal{W}_1 \cap \mathcal{S}_1$ and $\mathcal{W}_2 \cap \mathcal{S}_2$ (in bold), and some characteristic rhombus inequalities (in green). In the figure, we have $n=5$, $\mathcal{W}_1 = \{ t-1 = -i+j \}$ and $\mathcal{W}_2 = \{ |t+3| = i+j \}$.
  • Figure 4.1: On the left, we show $\Pi(0000,0235)/\mathbb{Z}\mathbbm{1}_4$, from Example \ref{['4Implies1']}. On the right, we show the symmetric functions from Example \ref{['2DoesNotImply1']}.
  • ...and 1 more figures

Theorems & Definitions (100)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 90 more