Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices
Abstract
For a polynomial F(t) = 1 + a1 t + a2 t2 +… + ad td with coefficients in a ℚ-algebra ℛ, we consider Schur-positivity of the symmetric function ΩF(x) = ∏i ≥ 1 F(xi). 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 Xinc(P)(x). Additionally, we prove Schur-positivity of certain sums of chromatic symmetric functions, refining recent work of Amdeberhan--Shareshian--Stanley. In the case when ℛ = ℚ[z1,z2,…,zn], we prove necessary and sufficient conditions for Schur-positivity in terms of symmetric function evaluations. In particular, we show for u, v ∈ ℝ≥ 0n, sλ(u) ≥ sλ(v) for all partitions λ if and only if mλ(u) ≥ mλ(v) for all partitions λ, and these inequalities are equivalent to u log-majorizing v. Using representation theory of GL(ℂn), we prove Schur-positivity of ΩF(x) when F(t) = det(I + t∑i = 1m zi vi vi*) for a collection of vectors v1,v2,…, vm ∈ ℂn. As an application for this result, we introduce the Kirchhoff symmetric function KG(x) of a loopless graph G and show that it it always Schur-positive.
Related
- Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices
- Total positivity and symmetric spaces
- Simultaneous diagonalization via congruence: Nondegenerate families of symmetric matrices
- Magic positivity for polar duals of pseudo-symmetric smooth Fano polytopes
- Isospectral Deformations of Diagonal Matrices by Symmetric Matrices in Dimension Four
BibTeX
@misc{siegl2026schurpositve,
title = {{Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices}},
author = {Isaiah Siegl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25036},
eprint = {2610.25036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25036},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices
AU - Isaiah Siegl
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25036
UR - https://arxiv.org/abs/2610.25036
AB - For a polynomial F(t) = 1 + a_1 t + a_2 t^2 +… + a_d t^d with coefficients in a ℚ-algebra ℛ, we consider Schur-positivity of the symmetric function Ω_F(x) = ∏_i ≥ 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)(x). Additionally, we prove Schur-positivity of certain sums of chromatic symmetric functions, refining recent work of Amdeberhan--Shareshian--Stanley. In the case when ℛ = ℚ[z_1,z_2,…,z_n], we prove necessary and sufficient conditions for Schur-positivity in terms of symmetric function evaluations. In particular, we show for u, v ∈ ℝ_≥ 0^n, s_λ(u) ≥ s_λ(v) for all partitions λ if and only if m_λ(u) ≥ m_λ(v) for all partitions λ, and these inequalities are equivalent to u log-majorizing v. Using representation theory of GL(ℂ^n), we prove Schur-positivity of Ω_F(x) when F(t) = det(I + t∑_i = 1^m z_i v_i v_i^*) for a collection of vectors v_1,v_2,…, v_m ∈ ℂ^n. As an application for this result, we introduce the Kirchhoff symmetric function K_G(x) of a loopless graph G and show that it it always Schur-positive.
ER -