Mukhin–Varchenko population–flag variety conjecture with Bruhat-cell decomposition

Let $\mathfrak g$ be a Kac–Moody algebra, let $\Lambda_1,\dots,\Lambda_n$ be dominant integral weights, let $z_1,\dots,z_n$ be points, and let $\boldsymbol b$ be symmetric parameters. A population is the family of Bethe ansatz solutions generated by the reproduction procedures from a solution. For a tuple of polynomials in a population, fix the degrees of its component polynomials. Let $^t\mathfrak g$ be the Langlands-dual Kac–Moody algebra. Mukhin–Varchenko population–flag variety conjecture. Every population associated to this data is an algebraic variety isomorphic to the flag variety of $^t\mathfrak g$, and the parts consisting of tuples with fixed polynomial degrees are isomorphic to Bruhat cells of that flag variety. This is the geometric conjecture describing populations and their degree strata; the supplied text gives no resolution status.