Pattern bounds for principal specializations of $\beta$-Grothendieck polynomials

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Hugh Dennin
September 27, 2022
3:00PM - 4:00PM
Location
MW 154

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Add to Calendar 2022-09-27 15:00:00 2022-09-27 16:00:00 Pattern bounds for principal specializations of $\beta$-Grothendieck polynomials 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/ MW 154 Department of Mathematics math@osu.edu America/New_York public
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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/

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