Title: Pattern bounds for principal specializations of $\beta$-Grothendieck polynomials
Speaker: Hugh Dennin (Ohio State)
Speaker's URL: https://math.osu.edu/people/dennin.3
Abstract: The principal specialization of a Schubert polynomial $\nu_w := \mathfrak{S}_w(1,\dots,1)$ is known to give the degree of the matrix Schubert variety corresponding to the permutation $w$. In this talk, we discuss a pattern containment formula (originally conjectured by Yibo Gao) which gives a lower bound for $\nu_w$ whenever $w$ is $1243$-avoiding. We then see how to extend this result to principal specializations of $\beta$-Grothendieck polynomials $\nu^{(\beta)}_w := \mathfrak{G}^{(\beta)}_w(1,\dots,1)$ in the setting where $w$ is additionally vexillary. Our methods are bijective, utilizing diagrams called bumpless pipe dreams to obtain combinatorial interpretations of the coefficients $c_w$ and $c^{(\beta)}_w$ appearing in these bounds.
URL associated with Seminar: https://research.math.osu.edu/agseminar/
