Georgia Institute of Technology College of Sciences

A symmetric ideal in the polynomial ring of a countable number of variables is an ideal that is invariant under any permutations of the variables. While such ideals are usually not finitely generated, Aschenbrenner and Hillar proved that such ideals are finitely generated if you are allowed to apply permutations to the generators, and in fact there is a notion of a Gröbner bases of these ideals. Brouwer and Draisma showed an algorithm for computing these Gröbner bases. Anton Leykin, Chris Hillar and I have implemented this algorithm in Macaulay2. Using these tools we are exploring the possible invariants of symmetric ideals that can be computed, and looking into possible applications of these algorithms, such as in graph theory.