Mathematics Colloquium
Speaker: Apoorva Khare (Indian Institute of Science)
Host: Dipendra Prasad
Title: Determinants with any smooth function reveal all Schur polynomials
Time, day and date: 4:00:00 PM, Wednesday, April 09
Venue: Ramanujan Hall
Abstract: Cauchy's identity (1840s) expands the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the $n \times n$ rank-one matrix $(u_i v_j)$. This was generalized by Frobenius (1880s). In a different century and context, Loewner (1960s) showed the vanishing of the initial Taylor coefficients of $\det f[t \cdot {\bf u}{\bf u}^T]$, where $f$ is a smooth function. This theme also appears recently in the 2010s in matrix analysis, for $f$ a polynomial.

This talk aims to bring this algebra and analysis together, by expanding $\det f[t \cdot {\bf u}{\bf v}^T]$ for all power series $f$. Time permitting, we will go from determinants to immanants for any character of the symmetric group, for bosonic/fermionic variables $u_i$ and $v_j$. (Partly based on joint works with Alexander Belton, Dominique Guillot and Mihai Putinar; with Siddhartha Sahi; and with Terence Tao.)