The weak Bruhat order on symmetric groups and the divergence of Schubert Polynomials

In this talk I will introduce the weak Bruhat order \(W_n\) on \(S_n\) and the strong Sperner property of Posets and show that \(W_n\) is strongly Sperner with approaches from two different works. Stanley conjectured an order rising operator \(U\) on \(\mathbb{C}W_n\) and showed that if \(U^{r-2k}\) is invertible then \(W_n\) is strongly Sperner. In the first work, Gaetz and Gao constructed an order lowering operator \(D\) which led to an \(sl_2\) representation of \(\mathbb{C}W_n\) and used representation theory of \(sl_2\) to prove invertibility. In the second work, Speyer et al. proved a determinantal formula of \(U^{r-2k}\) conjectured by Stanley, through showing that the divergence operator acts on Schubert Polynomials as the operator \(U\) and this implies invertibility of \(U^{r-2k}\).

Date published: Wednesday, March 15, 2023