Combinatorial reciprocity for symmetric functions

Question

I am wondering whether a certain instance of combinatorial reciprocity (in the sense of Stanley's classic paper "Combinatorial Reciprocity Theorems"), concerning symmetric functions, is known or follows from a more general result.

I use standard notion for symmetric functions as in e.g. Stanley's EC2.

For a symmetric function $f \in \Lambda$, and vector $\mathbf{x}=(x_1,\ldots,x_k)$ of fixed complex numbers $x_1,\ldots,x_k \in \mathbb{C}$, define $$\Phi(f,\mathbf{x};n) = f(\overbrace{x_1,\ldots,x_1}^{\textrm{$n$ times}},\overbrace{x_2,\ldots,x_2}^{\textrm{$n$ times}},\ldots,\overbrace{x_k,\ldots,x_k}^{\textrm{$n$ times}}).$$

It is clear that for power sum symmetric functions we have $\Phi(p_m,\mathbf{x};n) = n \cdot p_m(x_1,\ldots,x_k)$. Hence, since the $p_i$ generate $\Lambda$ as a ring, for any $f \in \Lambda$ we have that $\Phi(f,\mathbf{x};n)$ is a polynomial in $n$. So we can make sense of $\Phi(f,\mathbf{x};-n)$.

Proposition For any $f \in \Lambda$ of degree $m$, $\Phi(f,\mathbf{x};-n) = (-1)^{m} \cdot \Phi(\omega(f),\mathbf{x};n)$, where $\omega\colon \Lambda \to \Lambda$ is the canonical involution on the ring of symmetric functions swapping $e_n$ and $h_n$.

Proof: It suffices to check on a basis of $\Lambda$. For power sum symmetric functions we have $\Phi(p_{\lambda},\mathbf{x};n) = n^{\ell(\lambda)} p_{\lambda}(x_1,\ldots,x_k)$ and $\omega p_{\lambda} = (-1)^{|\lambda|-\ell(\lambda)} p_{\lambda}$, which together show that the proposition holds for the $p_{\lambda}$. $\square$

Using the standard non-intersecting paths interpretation of Schur functions, another, arguably more combinatorial proof of this proposition (showing $\Phi(s_{\lambda},\mathbf{x};-n)= (-1)^{|\lambda|}\Phi(s_{\lambda^t},\mathbf{x};n)$) can also be extracted from Gjergji Zaimi's beautiful answer to a previous MO question of mine, in which he establishes a general reciprocity theorem for non-intersecting paths in "growing" planar networks.

Question: Is this instance of combinatorial reciprocity known? Is it e.g. an example of Ehrhart-Macdonald reciprocity? Is there anything more interesting to be said about it?

Answer 1

Suppose we have two sets of variables, $\{x_i\}_{i\geq 1},\{y_i\}_{i\geq 1}$ and we evaluate the symmetric function $f$ with their pairwise products as inputs: $$f(x_1y_1,x_1y_2,...,x_iy_j,...).$$ Assume that $f$ decomposes as $f_0+f_1+f_2+\cdots$, with $f_n$ homogeneous of degree $n$. We can expand it as a symmetric function in the $y$ variables and obtain $$f(x_1y_1,x_1y_2,...,x_iy_j,...)=\sum_{n=0}^{\infty}\sum_{\lambda \vdash n}s_{\lambda}(y_1,y_2,\dots)f_{|\lambda|}\ast s_{\lambda}(x_1,x_2,\dots)$$ where $f\ast g$ denotes the internal product of symmetric functions (see for example pages 115-116 in Macdonald's book).

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In your situation, $\Phi(f,\mathbf{x};n)$ corresponds to specializing the $y$ vector to be $n$ copies of $1$ and the rest equal to $0$. Using the expansion above, we see that the reciprocity in question boils down to understanding the following fact:

$s_{\lambda}(1^n)$ is a polynomial in $n$ and when evaluated at a negative integer $n=-m$, it gives $(-1)^{|\lambda|} s_{\lambda^t}(1^m)$.

Algebraically this is apparent from the hook-content formula, but perhaps more relevant to you is the fact that we are enumerating SSYT's which fit in the framework of (P,w) partitions. This reciprocity then is a special case of Stanley's reciprocity theorem from the paper you originally linked.