Back-To-School seminar: Non-Intersecting Paths and Real-Rooted Polynomials

A matrix is totally non-negative (TNN) if all of the determinants of square submatrices are non-negative. Lindstrom's lemma gives a correspondence between TNN matrices and graphs that can be drawn in a disc, with minors of the matrix corresponding to collections of non-intersecting paths in the graph. I will show how Lindstrom's lemma can be used to give inequalities on the coefficients of polynomials with negative real roots. These inequalities then give Schur-positivity results for symmetric functions obtained from polynomials with negative real roots.