Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices

Abstract

For a polynomial $F(t) = 1 + a_1 t + a_2 t^2 +\dots + a_d t^d$ with coefficients in a $\mathbb{Q}$-algebra $\mathcal{R}$, we consider Schur-positivity of the symmetric function $Ω_F(\mathbf{x}) = \prod_{i \geq 1} F(x_i)$. Using work of Lindström and Gessel--Viennot, we give sufficient conditions for Schur-positivity in terms of directed networks. In the case when $F(t)$ is the multivariate independence polynomial of the incomparability graph $inc(P)$ of a natural unit interval order, we apply these ideas to give a new proof of Schur-positivity of the chromatic symmetric function $X_{inc(P)}(\mathbf{x})$. Additionally, we prove Schur-positivity of certain sums of chromatic symmetric functions, refining recent work of Amdeberhan--Shareshian--Stanley. In the case when $\mathcal{R} = \mathbb{Q}[z_1,z_2,\dots,z_n]$, we prove necessary and sufficient conditions for Schur-positivity in terms of symmetric function evaluations. In particular, we show for $\mathbf{u}, \mathbf{v} \in \mathbb{R}_{\geq 0}^n$, $s_λ(\mathbf{u}) \geq s_λ(\mathbf{v})$ for all partitions $λ$ if and only if $m_λ(\mathbf{u}) \geq m_λ(\mathbf{v})$ for all partitions $λ$, and these inequalities are equivalent to $\mathbf{u}$ log-majorizing $\mathbf{v}$. Using representation theory of $GL(\mathbb{C}^n)$, we prove Schur-positivity of $Ω_F(\mathbf{x})$ when $F(t) = \det(I + t\sum_{i = 1}^m z_i \mathbf{v}_i \mathbf{v}_i^*)$ for a collection of vectors $\mathbf{v}_1,\mathbf{v}_2,\dots, \mathbf{v}_m \in \mathbb{C}^n$. As an application for this result, we introduce the Kirchhoff symmetric function $K_G(\mathbf{x})$ of a loopless graph $G$ and show that it it always Schur-positive.