Algebra and Combinatorics Seminar: Apoorva Khare, Indian Institute of Science (Bangalore, India), Schur polynomials: from smooth functions to symmetric function identities
July 17 | 12:45 pm - 1:35 pm EDT
Cauchy’s determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix f[u v^T], where f(t) = 1/(1-t) is applied entrywise to the rank-one matrix u v^T = (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. The key novelty here has been an application of Schur polynomials, which essentially arise from a determinantal identity for p[u v^T].
I will explain the above journey from matrix positivity to determinantal identities and Schur polynomials; then go beyond, to the expansion for f[u v^T] for all power series f. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.)