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

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dc.creatorZabrocki, Mike
dc.date1999-01-05
dc.date1999-08-10
dc.date.accessioned2026-07-07T05:27:27Z
dc.date.available2026-07-07T05:27:27Z
dc.descriptionWe 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, $a_μ(T)$ and $b_μ(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\la$, $K_{\laμ}(q,t) = \sum_T t^{a_μ(T)} q^{b_μ(T)}$ for the case when when $μ$ 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 $μ$ is two columns.
dc.descriptionLaTEX, 37 pages, Replacement of submission with vertex operators only
dc.identifierhttps://arxiv.org/abs/math/9901016
dc.identifierhttp://arxiv.org/abs/math/9901016
dc.identifierElectron. J. Combinat. 6, R41 (1999), 36pp.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77925
dc.subjectCombinatorics
dc.subject05E10
dc.titlePositivity for special cases of $(q,t)$-Kostka coefficients and standard tableaux statistics
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