Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}\_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is *harmonic* if
$\varphi(\emptyset)=1$ and


\begin{equation}
\varphi(u)=\sum\_{\stackrel{\scriptstyle u \, \triangleleft \, v}{|v| \, = \, |u| + 1}} \varphi(v)
\end{equation}


where $\emptyset$ is the unique bottom element of $P$ (which we require to exist) and where $|u|$ denotes the rank of an element $u \in P$. The poset $P$ is 1-*differential* if in addition


$\bullet$ The number of elements covered by both $u$ and $v$ equals the number of elements in $P$ covering both $u$ and $v$ whenever $u \ne v$.


$\bullet$ If $u \in P$ covers exactly $k$ elements then $u$ is covered by exactly $k+1$ elements.


In (<https://arxiv.org/abs/math/9712266>) Goodman and Kerov introduced a semi-group flow on the space of
harmonic functions ${\frak{H}}(P)$ when $P$ is 1-differential.
Specifically, for $\tau \in [0,1]$ and $\varphi \in {\frak{H}}(P)$


\begin{equation}
C\_\tau(\varphi)(v)
\, := \, 
\sum\_{k=0}^{|v|} {\tau^k (1-\tau)^{|v|-k} \over {(|v| -k )! }}
\sum\_{|u| = k} \varphi(u) \dim(u,v)
\end{equation}


where $\dim(u,v)$ is the number of saturated chains
$p\_{|u|} \! \lhd \cdots \lhd p\_{|v|}$ in $P$
starting at $p\_{|u|} = u$ and ending at $p\_{|v|}= v$.
A fairly easy calculation reveals that $C\_\tau(\varphi)$
is harmonic whenever $\varphi$ is and that $C\_\tau (C\_\sigma(\varphi)) = C\_{\tau \sigma}(\varphi)$. Furthermore
we recover the original function $\varphi$
when $\tau = 1$ while we obtain the function


\begin{equation}
v \mapsto {1 \over {|v|!}} \dim(\emptyset, v) 
\end{equation}


when $\tau = 0$,
which is known to be harmonic whenever $P$ is 1-differential.


Now consider the Young lattice $(\Bbb{Y}, \subseteq)$
of all integer partitions, ordered by inclusion of their
respective Young diagrams. In virtue of the Pieri rule
we know that the function


\begin{equation}
\varphi(\lambda) \, := \, {s\_\lambda({\bf x}) \over {s^n\_{\Box}({\bf x})}} \quad
\text{where $\lambda \vdash n$}
\end{equation}


is a harmonic function on $\Bbb{Y}$ where
$s\_\lambda({\bf x})$ is the Schur function associated to
$\lambda$ and $s\_\Box({\bf x}) = x\_1 + x\_2 + x\_3 + \cdots < \infty$
is the Schur function associated to the partition $(1)$.
Let's apply the Goodman-Kerov flow to this
function: For $\lambda \vdash n$ we get


\begin{equation}
C\_\tau (\varphi)(\lambda) \, = \,
{1 \over {s^n\_\Box({\bf x})}} \, 
\underbrace{\sum\_{k=0}^{n} {\tau^k (1-\tau)^{n-k} \over {(n -k )! }} s\_\Box^{n-k}({\bf x})
\sum\_{|\mu| = k} s\_\mu({\bf x}) \dim(\mu,\lambda)}\_{\text{call this $s\_\lambda({\bf x};\tau)$}}
\end{equation}


We may now expand $s\_\lambda({\bf x}; \tau)$ as $\sum\_{\rho \vdash n} a\_{\lambda, \rho}(\tau) s\_\rho({\bf x})$.


**Question:** What can be said about the polynomials
$a\_{\lambda, \rho}(\tau)$? Have they already been identified/considered in the literature?


**Sub-question:** If the coefficient polynomials $a\_{\lambda, \rho}(\tau)$ are (in general) messy, does anything nice happen with $s\_\lambda({\bf x}; \tau)$ when we perform either the principal or content specializations, i.e.


\begin{equation}
\begin{array}{ll}
x\_i \mapsto \ \ q^{i-1} \ \ \text{for all $i \geq 1$} & \\ 
x\_i \mapsto 
\left\{
\begin{array}{ll} 1 &\text{for all $i \leq d$} \\
0 &\text{for all $i > d$}
\end{array} \right.
&\text{for some fixed but far out $d \geq 1$} 
\end{array}
\end{equation}


thanks, ines.



Harmonic flow on the Young lattice
Let us identify $s\_\lambda$ with the character of the irreducible representation $S^\lambda$ of the symmetric group $S\_n$ indexed by the partition $\lambda$. Then
$$
s\_\Box^{n-k}({\bf x})
\sum\_{|\mu| = k} s\_\mu({\bf x}) \dim(\mu,\lambda)
$$
defines the character of a certain representation of $S\_n$. We can explicitly describe it as follows. Since the Young lattice is the branching graph of representations of $S\_n$, $\dim(\mu,\lambda)$ is exactly the multiplicity of $S^\mu$ in the restriction of $S^\lambda$ from $S^n$ to $S^k$ (where $n = |\lambda|$ and $k = |\mu|$), so $\sum\_{|\mu| = k} s\_\mu({\bf x}) \dim(\mu,\lambda)$ is nothing but the character of the restriction of $S^\lambda$ to $S\_k$. Similarly, it is a well known fact that multiplication by $s\_\Box$ corresponds to induction from $S\_m$ to $S\_{m+1}$. So the quantity displayed above is the character of
$$
\mathrm{Ind}\_{S\_k}^{S\_n}\left( \mathrm{Res}\_{S\_k}^{S\_n}\left(S^\lambda \right)\right) = \mathrm{Ind}\_{S\_k}^{S\_n}(1) \otimes S^\lambda.
$$
So this is the same thing as tensoring $S^\lambda$ with the representation obtained by inducing the trivial representation from $S\_k$ to $S\_n$. The induced representation $\mathrm{Ind}\_{S\_k}^{S\_n}(1)$ that we are tensoring with might be more familiar under the notation $M^{(k, 1^{n-k})}$ (it is a permutation module). Its character expresses as $h\_k s\_\Box^{n-k}$, where $h\_k$ is the complete symmetric function of degree $k$.


Now we will use the internal product of symmetric functions defined by $p\_\mu \* p\_\nu = \delta\_{\mu, \nu} z\_\mu p\_\mu$. So in particular, homogeneous symmetric functions of different degrees multiply to zero. It is convenient for us because it describes the tensor product of representations of symmetric groups: $s\_\mu \* s\_\nu = \sum\_{\lambda} k\_{\mu, \nu}^\lambda s\_\lambda$, where $k\_{\mu, \nu}^\lambda$ is a Kronecker coefficient.


So now we may express
$$
s\_\lambda({\bf x}; \tau) = \sum\_{k=0}^{n} {\tau^k (1-\tau)^{n-k} \over {(n -k )! }} s\_\Box^{n-k}({\bf x})
\sum\_{|\mu| = k} s\_\mu({\bf x}) \dim(\mu,\lambda) = \left( \sum\_{k=0}^{n} {\tau^k (1-\tau)^{n-k} \over {(n -k )! }} h\_k s\_\Box^{n-k}\right) \* s\_\lambda.
$$
To tidy up this equation, we sum over all possible values of $n$ and $k$ (there is no harm in doing this because the introduced terms will become zero upon taking the internal product with $s\_\lambda$), which gives us a generating function
$$
\left(\left( \sum\_{k} \tau^k h\_k \right)\left( \sum\_l \frac{(1-\tau)^l s\_\Box^{l}}{l!}\right)\right) \* s\_\lambda = \left(H(\tau) \exp((1-\tau)s\_\Box) \right) \* s\_\lambda,
$$
where $l = n-k$, and $H(\tau) = \sum\_{k \geq 0} \tau^k h\_k$ is the generating function of complete symmetric functions. Now, we can express $H(\tau)$ in terms of power-sum symmetric functions as
$$
H(\tau) = \exp\left( \sum\_{i \geq 1} \frac{p\_i \tau^i}{i} \right).
$$
Using the fact that $s\_\Box = p\_1$, we are left with
$$
s\_\lambda({\bf x}; \tau) = \exp\left( p\_1 + \sum\_{i \geq 2} \frac{p\_i \tau^i}{i} \right) \* s\_\lambda = \sum\_{|\mu| = n} \frac{\chi\_\mu^\lambda p\_\mu}{z\_\mu} \tau^{n-m\_1(\mu)},
$$
where $m\_1(\mu)$ is the number of parts of size 1 in $\mu$ and we used the equation $s\_\lambda = \sum\_\mu \frac{\chi\_\mu^\lambda p\_\mu}{z\_\mu}$.


But the quantity that was asked about, $a\_{\lambda, \rho}(\tau)$, is the coefficient of $s\_\rho$ in this expression:
$$
a\_{\lambda, \rho}(\tau) = \langle s\_\rho, \exp\left( p\_1 + \sum\_{i \geq 2} \frac{p\_i \tau^i}{i} \right) \* s\_\lambda \rangle = \sum\_{|\mu| = n} \frac{\chi\_\mu^\rho \chi\_\mu^\lambda}{z\_\mu} \tau^{n - m\_1(\mu)},
$$
So this is like the inner product between two Schur functions with respect to a deformed inner product where power sums $p\_i$ are weighted by a factor of $\tau$ if $i \geq 2$.


As a function of $\tau$, this interpolates between the case $\tau = 1$, where we just get $\langle s\_\rho, s\_\lambda \rangle = \delta\_{\rho, \lambda}$, and the case where $\tau = 0$ (where only $\mu = (1^n)$ has a nonzero contribution to the sum) which gives $\frac{\dim(\rho)\dim(\lambda)}{n!}$. I do not recall seeing this particular interpolation before.


Regarding the evaluations you mention, they can be described using homomorphisms $\varphi: \Lambda \to \mathbb{C}$ defined by
$$
\varphi(p\_n) = 1^n + q^n + q^{2n} + \cdots = (1-q^n)^{-1}
$$
(or $\varphi(p\_n) = d$ in the latter case). The upshot here is that we can describe the result of applying $\varphi$ to $s\_\lambda({\bf x}; \tau)$ as the result of applying a different homomorphism to the Schur function $s\_\lambda$. Specifically, $\varphi(s\_\lambda({\bf x}; \tau)) = \psi(s\_\lambda)$, where
$$
\psi(p\_n) = \frac{\tau^n}{1-q^n}
$$
for $n \geq 2$ and $\psi(p\_1) = (1-q)^{-1}$. (In the latter case we take $\psi(p\_n) = d\tau^n$ for $n\geq 2$ and $\psi(p\_1) = d$.) The fact that these descriptions are not uniform (have a different case for $n=1$) makes me suspect there might not be nice formulas for the evaluations.


