Symmetric Ideals

Abstract

Polynomials appear in many different fields such as statistics, physics and optimization. However, when the degrees or the number of variables are high, it generally becomes quite difficult to solve polynomials or to optimize polynomial functions. An approach that can often be helpful to reduce the complexity of such problems is to study symmetries in the problems. A relatively new field, that has gained a lot of traction in the last fifteen years, is the study of symmetry in polynomial rings in increasingly many variables. By considering the action of the symmetric groups on these polynomial rings, one can for instance show that certain sequences of symmetric ideals in increasingly larger polynomial rings are finitely generated up to orbits.

In this thesis we will investigate some properties of such sequences. In particular the Hilbert Series and Gröbner bases of Specht ideals, a class of ideals arising from the representation theory of the symmetric group. We prove a conjectured Gröbner basis for Specht ideals of shape (n−k, 1^k) and give two different criteria for verifying the conjecture for other Specht ideals. We also build on a result from the representation theory of the symmetric group by showing that the leading monomials of the standard Specht polynomials span the vector space of leading monomials of Specht polynomials.