A Macdonald Vertex Operator and Standard Tableaux Statistics

The two parameter family of coefficients Kλμ(q, t) introduced by Macdonald are conjectured to (q, t) count the standard tableaux of shape λ. If this conjecture is correct, then there exist statistics aμ(T ) and bμ(T ) such that the family of symmetric functions Hμ[X; q, t] = ∑ λKλμ(q, t)sλ[X] are generating functions for the standard tableaux of size |μ| in the sense that Hμ[X; q, t] = ∑ T q aμ(T )tbμ(T sλ(T )[X] where the sum is over standard tableau of of size |μ|. We give a formula for a symmetric function operator H 2 with the property that H qt 2 H(2a1b)[X; q, t] = H(2a+11b)[X; q, t]. This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that H(2a1b)[X; q, t] is the generating function for standard tableaux of size 2a + b (and hence that Kλ(2a1b)(q, t) is a polynomial with non-negative integer coefficients). The inductive proof gives an algorithm for ’building’ the standard tableaux of size n + 2 from the standard tableaux of size n and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics aμ(T ) and bμ(T ) when μ is of the form (21) and that these statistics prove conjectures about the relationship between adjacent rows of the (q, t)-Kostka matrix that were suggested by Lynne Butler.