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Positivity for special cases of $$(q,t)$$-Kostka coefficients and standard tableaux statistics. (English) Zbl 0931.05086
Electron. J. Comb. 6, No. 1, Research paper R41, 36 p. (1999); printed version J. Comb. 6, 531-566 (1999).
We present two symmetric function operators $$H_3^{qt}$$ and $$H_4^{qt}$$ that have the following properties: $$H_{m}^{qt} H_{(2^a1^b)}(X;q,t) = H_{(m2^a1^b)}(X;q,t)$$ 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 $$\lambda$$, $$K_{\lambda\mu}(q,t) = \sum_T t^{a_{\mu}(T)} q^{b_{\mu}(T)}$$ for the case when $$\mu$$ is two columns or of the form $$(32^a1^b)$$ or $$(42^a1^b)$$. This provides a proof of the positivity of the $$(q,t)$$-Kostka coefficients in the previously unknown cases of $$K_{\lambda (32^a1^b)}(q,t)$$ and $$K_{\lambda (42^a1^b)}(q,t)$$. 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.

##### MSC:
 5e+10 Combinatorial aspects of representation theory
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