Some Schubert shenanigans

Abstract

We give a conjectured evaluation of the determinant of a certain matrix $\tilde{D}(n,k)$. The entries of $\tilde{D}(n,k)$ are either 0 or specializations $\mathfrak{S}_w(1,\dots,1)$ of Schubert polynomials. The conjecture implies that the weak order of the symmetric group $S_n$ has the strong Sperner property. A number of peripheral results and problems are also discussed.