Combinatorics Seminar

Speaker: Ayah Almousa

Title: Polarizations of Powers of Graded Maximal Ideals

Abstract: A "polarization" of a monomial ideal is a square-free monomial ideal in a larger polynomial ring which preserves important homological invariants. Many commutative algebraists are familiar with the use of the "standard" polarization, but the first use of a nonstandard polarization was by Nagel and Reiner in the 2000s, who used the "box polarization" to produce a minimal cellular resolution for strongly stable ideals. This leads to the natural question: what other ways are there to polarize a monomial ideal, and what other applications might there be for these non-standard polarizations? In the first half of this talk, I will give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal in a polynomial ring. In the second half, I will give a combinatorial description of their Alexander duals and discuss applications of polarizations to commutative algebra, algebraic geometry, and combinatorics. This is joint work with Gunnar Fløystad and Henning Lohne