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

For a polynomial $F(t)=1+a_{1}t+a_2{t}^2+\cdots +a_d{t}^d$ with coefficients in a $\mathbb{Q}$-algebra $\mathcal{R}$, we consider Schur-positivity of the symmetric function ${\mathrm{\Omega }}_F(\mathbf{x})={\prod }_{i\ge 1}F(x_i)$. 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 $\mathbf{u},\mathbf{v}\in {\mathbb{R}}_{\ge 0}^n$, $s_{\lambda }(\mathbf{u})\ge s_{\lambda }(\mathbf{v})$ for all partitions $\lambda$ if and only if $m_{\lambda }(\mathbf{u})\ge m_{\lambda }(\mathbf{v})$ for all partitions $\lambda$, 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 ${\mathrm{\Omega }}_F(\mathbf{x})$ when $F(t)=\det (I+t{\sum }_{i=1}^m{z}_i{\mathbf{v}}_i{\mathbf{v}}_i^{\ast })$ 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.