Combinatorial definition of Hall–Littlewood polynomials (sum over SSYT?)

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.

• 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? Commented Mar 17, 2015 at 10:20
• Pardon my French. Commented Mar 17, 2015 at 10:35
• @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). Commented Mar 21, 2015 at 21:27
• @IgorMakhlin (И про t-версию Бриона и т.п. мы бы с М.Б. с интересом послушали в какой-то момент.) Commented Mar 21, 2015 at 21:28
• 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 Commented Mar 22, 2015 at 8:42