Brendon Rhoades

A polynomial in the ringQ[x1, . . . , xn] is calledsymmetricif it is invariant under any permutation of the vari-ablesx1, . . . , xn. We will overview various properties of the sub-ring of symmetric polynomials, the ‘invariant ideal’Ingener-ated by symmetric polynomials with vanishing constant term,and the quotientRnof the polynomial ringQ[x1, . . . , xn] byIn.Thecoinvariant algebraRnhas algebraic properties which aredeeply tied to the combinatorics of permutations in the sym-metric groupSn. Time permitting, we will discuss a new gen-eralizationRn,kof the coinvariant algebra whose properties aresimilarly related to the combinatorics ofordered set partitions.