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Hall–Littlewood polynomials $P_\lambda(x;t)$ is an important deformation of Schur polynomials forming a basis in the ring of symmetric polynomials over $\mathbb Z[t]$. There are various definitions, including quite explicit $$ P_\lambda(x;t)= \sum_{w\in S_n/\lambda} w\left(x^\lambda\prod_{\lambda_i>\lambda_j}\frac{x_i-tx_j}{x_i-x_j}\right) $$ but I’m interested in a combinatorial description.

Schur polynomial $s_\lambda$ is a sum of monomials corresponding to semi-standard Young tableux of shape $\lambda$ — so I'm expecting an answer in the form of a weight on SSYT.

P.S. I'm mostly interested in the principal specialization of H-L polynomials — maybe this $(q,t)$-weight on SSYT is easier to describe than the full answer.

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  • $\begingroup$ Наткнулся на твой вопрос и решил уж было, что настал мой звездный час, но потом понял, что за Grigory M. Правильный ответ тебе Миша уже рассказал наверняка, да? $\endgroup$
    – imakhlin
    Commented Mar 17, 2015 at 10:20
  • $\begingroup$ Pardon my French. $\endgroup$
    – imakhlin
    Commented Mar 17, 2015 at 10:35
  • $\begingroup$ @IgorMakhlin О, привет. Well, yes and no: there is an explicit description of the $t$-weight $\psi$ in Macdonald's book — but it's complicated and not terribly satisfying. So if you have a better answer, please explain it (here or iRL). $\endgroup$
    – Grigory M
    Commented Mar 21, 2015 at 21:27
  • $\begingroup$ @IgorMakhlin (И про t-версию Бриона и т.п. мы бы с М.Б. с интересом послушали в какой-то момент.) $\endgroup$
    – Grigory M
    Commented Mar 21, 2015 at 21:28
  • $\begingroup$ Apparently, I do not have a better answer, although I've spent quite some time trying to come up with some more insightful perspective: mathoverflow.net/questions/199629 $\endgroup$
    – imakhlin
    Commented Mar 22, 2015 at 8:42

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I'm not sure if this is exactly what you're looking for, but I recently worked on a paper regarding just this question (http://arxiv.org/abs/1403.8139). We were generalizing Tokuyama's deformation formula (http://projecteuclid.org/euclid.jmsj/1230129805) for the Schur polynomial, so we used Gelfand-Tsetlin patterns rather than Young tableaux. Tokuyama's original formula is in terms of strict GT patterns, which are in bijection with standard Young tableaux, but extending to the Hall-Littlewood polynomials required using nonstrict GT patterns (which are in bijection with SSYT).

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  • $\begingroup$ To me GZ-patterns are as good as SSYT (the bijection is straightforward anyway) — thank you, I'll take a look! $\endgroup$
    – Grigory M
    Commented Jan 1, 2015 at 17:37
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It is also possible to get a combinatorial description using semi-standard augmented fillings, by specializing the combinatorial formula for Macdonald polynomials (see wikipedia on the Macdonald polynomials).

It is also possible to get a formula as sum over SSYT's and the cocharge statistic. This gives the expansion of HL-polys in terms of Schur polynomials.

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  • $\begingroup$ Wikipedia page on Macdonald polynomials a) doesn't contain words 'semi-standard' or 'augmented'; b) doesn't give a combinatorial description of Macdonald polynomials — it mentions a description of transformed Macdonald polynomials that uses 'certain combinatorial statistics' inv and maj. I guess details can be found in arxiv.org/abs/math/0409538 $\endgroup$
    – Grigory M
    Commented May 6, 2015 at 18:55
  • $\begingroup$ ...but extracting something explicit for HL polynomials from that doesn't look like an easy task... $\endgroup$
    – Grigory M
    Commented May 6, 2015 at 18:55
  • $\begingroup$ It depends what you mean by explicit; You can easily write this as a sum over a family of tableaux, with some not-so-difficult combinatorial statistics on said tableaux. $\endgroup$
    – Per Alexandersson
    Commented May 6, 2015 at 19:04

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