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Consider the Hall-Littlewood polynomial $$ P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-tx_j}{x_i-x_j}\right), $$ where $\lambda=(\lambda_1,\ldots,\lambda_n)$ is a partition and $S_n^\lambda$ is the stabilizer of $\lambda$. These give a $\mathbb{Z}[t]$-basis for the ring of symmetric functions (with coefficients in $\mathbb{Z}[t]$). In particular, if we apply the involution $$t\mapsto -t,$$ we get such a symmetric polynomial, $P_\lambda(x_1,\ldots,x_n;-t)$, which we can expand as a linear combination of Hall-Littlewood polynomials: ie. there are unique polynomials $h_{\lambda,\mu}(t)$ such that $$ P_\lambda(x_1,\ldots,x_n;-t)=\sum_{\mu}h_{\lambda,\mu}(t)P_\mu(x_1,\ldots,x_n;t). $$

Is there a known expression for the coefficients $h_{\lambda,\mu}(t)$?

A couple things to say: since $P_\lambda(x;0)=s_\lambda(x)$ is the Schur polynomial, we need $h_{\lambda,\mu}(0)=\delta_{\lambda,\mu}$. For example, when $n=2$, it is simple to compute that $$ P_{(\lambda_1,\lambda_2)}(x_1,x_2;-t)=P_{(\lambda_1,\lambda_2)}(x_1,x_2;t)+\sum_{k=1}^{[\lambda_1-\lambda_2/2]}(2t^k)P_{(\lambda_1-k,\lambda_2+k)}(x_1,x_2;t), $$ where $[n]$ is the floor function. This is clearly a root string, so I am hoping there is a known expression (say in terms of tableaux or something) in general.

A second, vaguer question is

is there is a theoretic interpretation to the involution $t\mapsto -t$ in relation to these polynomials and their generalizations? By theoretic, I am referring to the myriad ways in which HL polynomials appear (in terms of Hecke algebras or geometric representation theory).

This question arose from certain computations with $p$-adic groups, and this old question seems to indicate that there may be something interesting to say.

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The transition matrix from the Schur functions to the HL symmetric functions is $K(t)$, the matrix of Kostka polynomials. This means that the transition matrix from $P(x;t)$ to $P(x;-t)$ is $K(t)^{-1}K(-t)$. This is upper-triangular with respect to the dominance partial order on partitions (or lower-triangular, depending on how you look at it), so $h_{\lambda\mu}(t) = 0$ unless $\lambda \succeq \mu$.

The character theory of $\text{GL}_n(\mathbb{F}_q)$ and $\text{U}_n(\mathbb{F}_{q^2})$ (the finite unitary group) can both be described in terms of symmetric functions, with the involution $q \leftrightarrow -q$ often relating the two structures. I don't know of any particularly meaningful interpretation of the HL symmetric functions in terms of characters of these groups, however.

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  • $\begingroup$ Thank you for the response! I am aware of the upper triangularity, but hoped for a more exact formulation. I find the character theory comment very interesting; is there a good reference? $\endgroup$ – WSL Sep 17 at 18:54
  • $\begingroup$ A good reference for the character theory of the finite general linear groups is "Hall Functions and Symmetric Polynomials," by MacDonald, which I'm guessing you're familiar with. Thiem and Vinroot describe the analogous construction for the finite unitary groups in "On the characteristic map of finite unitary groups." $\endgroup$ – Scott Andrews Sep 19 at 0:30

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