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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.