  We introduce and study symmetric polynomials, which as very special cases
include polynomials related to the supersymmetric eight-vertex model, and other
elliptic lattice models with $\Delta=\pm 1/2$. There is also a close relation
to affine Lie algebra characters. After a natural change of variables, our
polynomials satisfy a non-stationary Schr\"odinger equation with elliptic
potential, which is related to the Knizhnik-Zamolodchikov-Bernard equation and
to the canonical quantization of Painlev\'e VI. Moreover, specializations of
our polynomials can be identified with tau functions of Painlev\'e VI, obtained
from one of Picard's algebraic solutions by acting with a four-dimensional
lattice of B\"acklund transformations. In the present work, our results on
these topics are summarized with a minimum of technical details.
