Combinatorics Seminar

Consider the action of the symmetric group S_n on the polynomial ring Q[x_1, ..., x_n] by variable permutation. The coinvariant algebra R_n is the graded S_n-module obtained by modding out Q[x_1, \dots, x_n] by the ideal generated by 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 S_n-modules R_{n,k} 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 "Delta Conjecture" in the theory of Macdonald polynomials. Joint with Jim Haglund and Mark Shimozono. Speaker(s): Brendon Rhoades (UC San Diego)