Eric Dannetun: Principal symmetric ideals and their Hilbert functions

We introduce symmetric ideals, which are homogeneous ideals of polynomials rings that are closed under an action of the symmetric group. For any homogeneous ideal one has the property that there exists a monomial ideal which has the same Hilbert function, and a natural question is therefore whether for any symmetric ideal one can find a monomial symmetric ideal which has the same Hilbert function. We outline a proof showing that in general this is not the case. Lastly we give a complete description of this property in the case of 2 variables.