Cauchy’s determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the rank-one matrix $(u_i v_j)$. This theme has resurfaced in the 2010s in analysis (following a 1960s computation by Loewner), in the quest to find polynomials $p(t)$ with a negative coefficient that entrywise preserve positivity. A key novelty here has been the application of Schur polynomials, which essentially arise from the expansion of $\det(p[{\bf u}{\bf v}^T])$, to positivity.

In the first half of the talk, I will explain the above journey from matrix positivity to determinantal identities and Schur polynomials; then go beyond, to the expansion of $\det(f[{\bf u}{\bf v}^T])$ for all power series $f$. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.) In the second half, joint with Siddhartha Sahi, I will explain how to extend the above determinantal identities to (a) any subgroup $G$ of signed permutations; (b) any character of $G$, or even complex class function; (c) any commutative ground ring $R$; and (d) any power series over $R$.