UC San Diego

Jim Haglund, Jennifer Morse, and Mike Zabrocki have published papers introducing symmetric function operators, C operator and B operator, and their combinatorial interpretations and identities. Their compositional refinement of the Shuffle Conjecture can be used to represent the image of a schur function under the Bergeron -Garsia nabla operator as a weighted sum of a suitable collection of parking functions. First, we express a schur function under nabla operator with C operator using its definition and fundamental identities about symmetric functions. Then, the compositional Shuffle conjecture can be used as the expression of a family of [Triangle down]Cp₁ ··· Cpk1, with p = (p₁,··· ,pk), composition of n, in terms of a sum of weights of a suitable set of parking functions. Therefore, in this paper, we express the image of a schur function under nable operator as the difference x between two sums of the weights of certain sets of parking functions. Then, we introduce injections between those sets of parking functions for a few cases of schur functions, so that an image of schur functions under nabla operator can be simply expressed using the complementary set of parking functions. The validity of these expressions is still conjectural until the compositional refinement of the Shuffle Conjecture is proved