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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:
05E10 Combinatorial aspects of representation theory
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