Home /  Fellowship of the Ring, National Seminar: Betti numbers of monomial ideals fixed by permutations of the variables

Seminar

Fellowship of the Ring, National Seminar: Betti numbers of monomial ideals fixed by permutations of the variables March 25, 2021 (03:00 PM PDT - 04:30 PM PDT)
Parent Program: --
Location: SLMath: Online/Virtual
Speaker(s) Satoshi Murai
Description

To attend this seminar, you must register in advance, by clicking HERE.

Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Betti Numbers of Monomial Ideals Fixed by Permutations of the Variables.mp4

Abstract/Media

To attend this seminar, you must register in advance, by clicking HERE.

Let R_n be the polynomial ring with n variables over a field K. We consider the natural action of the n-th symmetric group S_n to R_n. In this talk, I will mainly talk about the following problem: Fix monomials u_1,\dots,u_m and consider the ideal I_n of R_n generated by the S_n-orbits of these monomials. How the Betti numbers of I_n change when n increases?

I will explain that there is a simple way to determine non-zero positions of the Betti table of I_n when n is sufficiently large. I also explain that we can determine the Betti numbers of I_n by considering the S_n-module structure of Tor_i(I_n,K).

The above problem is motivated by recent studies of algebraic properties of S_n-invariant ideals and is inspired by studies of Noetherianity up to symmetry. I will explain this motivation and basic combinatorial properties of S_n-invariant ideals in the first part of the talk.

This talk includes a joint work with Claudiu Raicu.

Asset no preview Notes 1 4.6 MB application/pdf
Asset no preview Notes 2 6.9 MB application/pdf

Betti Numbers of Monomial Ideals Fixed by Permutations of the Variables.mp4