An ideal in a polynomial ring is called symmetric if it is invariant under the natural action of the symmetric group by permutation of the variables. Among all symmetric monomial ideals, the class of symmetric shifted ideals plays an analogous role as the class of stable ideals among all monomial ideals, for instance, as they admit a componentwise linear resolution. In this talk, I will describe how the class of symmetric shifted ideals and its subclass of symmetric strongly shifted ideals behaves under most ideal operations, and in particular under taking powers. As an application, I will discuss our current understanding of the Rees algebra of a symmetric strongly shifted ideal $I$, which is obtained by combining together all the powers of $I$. This is part of ongoing joint work with Alexandra Seceleanu, sponsored by the 2021 AWM Mentoring Travel Grant.
Symmetric shifted ideals and their Rees algbera
Alessandra Costantini, Oklahoma State UniversityAuthors: Alessandra Costantini and Alexandra Seceleanu
2022 AWM Research Symposium
Combinatorial and Homological Methods in Commutative Algebra