Geodesic
Schur Polynomials, Entrywise Positivity Preservers, and Weak Majorization
Séminaire lotharingien de combinatoire, 80B (2018)

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We prove a monotonicity phenomenon for ratios of Schur polynomials. In this we are motivated by - and apply our result to - understanding polynomials and power series that preserve positive semidefiniteness (psd) when applied entrywise to psd matrices. We then extend these results to classify polynomial preservers of total positivity. As a further application, we extend a conjecture of Cuttler, Greene, and Skandera (2011) to obtain a novel characterization of weak majorization using Schur polynomials. Our proofs proceed through a Schur positivity result of Lam, Postnikov, and Pylyavskyy (2007), and computing the leading terms of Schur polynomials. 

@article{SLC_2018_80B_a13,
     author = {Apoorva Khare and Terence Tao},
     title = {Schur {Polynomials,} {Entrywise} {Positivity} {Preservers,} and {Weak} {Majorization}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a13/}
}
TY  - JOUR
AU  - Apoorva Khare
AU  - Terence Tao
TI  - Schur Polynomials, Entrywise Positivity Preservers, and Weak Majorization
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a13/
ID  - SLC_2018_80B_a13
ER  - 
%0 Journal Article
%A Apoorva Khare
%A Terence Tao
%T Schur Polynomials, Entrywise Positivity Preservers, and Weak Majorization
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a13/
%F SLC_2018_80B_a13
Apoorva Khare; Terence Tao. Schur Polynomials, Entrywise Positivity Preservers, and Weak Majorization. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a13/