Inversion of matrices, a $\C^*$ action on Grassmannians and the space of complete quadrics

Abstract: Let $\Gamma$ be the closure of the set of pairs $(A,A^{-1})$ of symmetric matrices of size $n$. In other words, $\Gamma$ is the graph of the inversion map on the space $\mathrm{Sym}_n$ of symmetric matrices of size $n$. What is the cohomology class of $\Gamma$ in the product of projective spaces? Equivalently, what is the degree of the variety $L^{-1}$ obtained as the closure of the set of inverses of matrices from a generic linear subspace $L$ of $\mathrm{Sym}_n$. This question is interesting in its own right but it is also motivated by algebraic statistics. In my talk, I will explain how to invert a matrix using a $\C^*$ action on Grassmannians, relate the above question to classical enumerative problems about quadrics, and give several possible answers.

This is joint work in progress with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, Andrzej Weber, and Jaroslaw A. Wisniewski.

algebraic geometrycombinatorics

Audience: researchers in the topic

( video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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