Mike Zabrocki - Special cases of positivity for (q,t)-Kostka coefficients

We present two symmetric function operators H₃^qt and H₄^qtthat have the property H₃^qt H_(2^a1^b)[X;q,t] = H_(32^a1^b)[X;q,t] and H₄^qt H_(2^a1^b)[X;q,t] = H_(42^a1^b)[X;q,t]. These operators are generalizations of the analogous operator H₂^qt and have expressions in terms of Hall-Littlewood vertex operators. The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when is two columns. This gives statistics, and , on standard tableaux such that the q,t Kostka polynomials are given by the sum over standard tableaux of shape , for the case when when is two columns or of the form (32^a1^b) or (42^a1^b). This provides proof of the positivity of the (q,t)-Kostka coefficients in the previously unknown cases of and .