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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.

05E10 Combinatorial aspects of representation theory
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