Positivity for special cases of $(q,t)$-Kostka coefficients and standard tableaux statistics

Abstract

We present two symmetric function operators $H_3^{qt}$ and $H_4^{qt}$ that have the property $H_{3}^{qt} H_{(2^a1^b)}[X;q,t] = H_{(32^a1^b)}[X;q,t]$ and $H_4^{qt} H_{(2^a1^b)}[X;q,t] = H_{(42^a1^b)}[X;q,t]$. These operators are generalizations of the analogous operator $H_2^{qt}$ and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, $a_{\mu}(T)$ and $b_{\mu}(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\la$, $K_{\la\mu}(q,t) = \sum_T t^{a_{\mu}(T)} q^{b_{\mu}(T)}$ for the case when when $\mu$ 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 $K_{\la (32^a1^b)}(q,t)$ and $K_{\la (42^a1^b)}(q,t)$. The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when $\mu$ is two columns.