Critical points of master functions and integrable hierarchies

Abstract

We consider the population of critical points generated from the trivial critical point of the master function with no variables and associated with the trivial representation of the affine Lie algebra $\hat{\frak{sl}}_N$. We show that the critical points of this population define rational solutions of the equations of the mKdV hierarchy associated with $\hat{\frak{sl}}_N$. We also construct critical points from suitable $N$-tuples of tau-functions. The construction is based on a Wronskian identity for tau-functions. In particular, we construct critical points from suitable $N$-tuples of Schur polynomials and prove a Wronskian identity for Schur polynomials.