On a positivity property of Hall-Littlewood polynomials

Question

Here's the new, more thought through version.

Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is dominant, in terms of $\mathfrak{gl}_n$ weights). Next, consider the polynomial $$P_\lambda(x_1,\ldots,x_n;t)=\sum\limits_{\sigma\in S_n} \sigma\left(x_1^{\lambda_1}\ldots x_n^{\lambda_n}\prod\limits_{i<j}\dfrac{x_i-tx_j}{x_i-x_j}\right).$$ This can, probably, be rightfully referred to as the Hall-Littlewood polynomial (all parts are distinct, so no normalization is needed). I'm just trying to accentuate the fact that I'm concerned with this specific polynomial, not an element of $\Lambda_{\mathbb{C}[t]}$. (Or is then "Hall-Littlewood polynomial" not the appropriate term?)

Anyway, I strongly believe that with our assumption on $\lambda$ in place the polynomial $P_\lambda(x_1,\ldots,x_n;-t)$ has positive coefficients. As I mentioned in the first version, this is confirmed by my observations and a certain geometrical argument.

My assumption can be somewhat weakened, but even as is this looks to me like a very basic fact in the theory of these well-studied expressions. Now, my questions are:

1) Can someone confirm that this is true and provide a reference to some down-to-earth (combinatorial) proof?

2) Why is this absent from all (almost all?) surveys on the subject of Hall-Littlewood polynomials? Just because this is a statement about the polynomials themselves rather than symmetric functions? Is it really not mentioned in Macdonald's book?

3) My real question. Is there a proof expressing $P_\lambda(x_1,\ldots,x_n;-t)$ as a sum of visibly positive summands enumerated by some combinatorial set (hopefully, SSYTs or Gelfand-Tsetlin patterns)?

Answer 1

There is a nice formula for Hall-Littlewood polynomials that follows from the combinatorial formula due to Haglund, Haiman and Loehr:

In this paper, eqution (69), we have that the Macdonald P-polynomial is given as $$ P_\lambda(x;q,t)= \left[\prod_{u \in dg'(\lambda)} (1-q^{l(u)+1}t^{a(u)})\right] \sum_{\gamma \sim \lambda} \frac{E_\gamma(x;1/q,1/t)}{\prod_{s\in \gamma}(1-q^{l(s)+1}t^{a(s)})} $$ where the sum is over all permutations (as compositions) of $\lambda$.

Putting $q=0$ then gives an expression for the Hall-Littlewood polynomials, See eq. (7.8) in this paper.

Thus, $$ P_\lambda(x;t) = \sum_{\gamma \sim \lambda} \sum_{F \in NAF(\gamma)} x^F t^{coinv(F)} (1-t)^{dn(F)} $$ where $dn(F)$ is the number of boxes $u$ in the filling $F$, such that the box to the left of $u$ is filled with a different entry from $u$. There are some details to be filled in, regarding a 0th column, that is referred to as the basement. Here, $NAF(\gamma)$ is a certain set of non-attacking fillings with weakly decreasing rows, and the 0th column has an $i$ in row $i$.

So, $$ P_\lambda(x;-t) = \sum_{\gamma \sim \lambda} \sum_{F \in NAF(\gamma)} x^F (-t)^{coinv(F)} (1+t)^{dn(F)} $$ It is tempting to hope that each inner sum is positive, but that is not the case. However, it should not be too tricky to find a sign-reversing involution to cancel negative signs. One can for example apply certain operators that preserve symmetric functions, but does something predictable on the $E_\gamma$, see this article, (apologizes for self-referencing) which also has some other similar-looking expressions for the Hall-Littlewood polynomials.

Answer 2

The specialization $P_\lambda(x;-1)$ is what is referred to as Schur's P functions. They can be described combinatorially using shifted tableaux, and are Schur-positive.

See also slides here by S. Cho.