Combinatorics Seminar

Consider the action of the polynomial ring $\mathfrak{S}_n$ on the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The {\em coinvariant algebra} $R_n$ is the graded $\mathfrak{S}_n$-module obtained by modding out $\mathbb{Q}[x_1, \dots, x_n]$ by the ideal generated by $\mathfrak{S}_n$-invariant polynomials with vanishing constant term. The algebraic properties of $R_n$ are governed by the combinatorial properties of permutations. We will introduce and study a family of graded $\mathfrak{S}_n$-modules $R_{n,k}$ which depend on two positive integers $k \leq n$ which reduce to the coinvariant algebra when $k = n$. The algebraic properties of the $R_{n,k}$ are governed by ordered set partitions of $\{1, 2, \dots, n\}$ with $k$ blocks. We will generalize results of E. Artin, Garsia-Stanton, Chevalley, and Lusztig-Stanley from $R_n$ to $R_{n,k}$. The modules $R_{n,k}$ are related to the {\em Delta Conjecture} in the theory of Macdonald polynomials. Joint with Jim Haglund and Mark Shimozono.