# Harmonic flow on the Young lattice

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

$$$$\varphi(u)=\sum_{\stackrel{\scriptstyle u \, \triangleleft \, v}{|v| \, = \, |u| + 1}} \varphi(v)$$$$

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 $$\varphi \in {\frak{H}}(P)$$

$$$$C_\tau(\varphi)(v) \, := \, \sum_{k=0}^{|v|} {\tau^k (1-\tau)^{|v|-k} \over {(|v| -k )! }} \sum_{|u| = k} \varphi(u) \dim(u,v)$$$$

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

$$$$v \mapsto {1 \over {|v|!}} \dim(\emptyset, v)$$$$

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

$$$$\varphi(\lambda) \, := \, {s_\lambda({\bf x}) \over {s^n_{\Box}({\bf x})}} \quad \text{where \lambda \vdash n}$$$$

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

$$$$C_\tau (\varphi)(\lambda) \, = \, {1 \over {s^n_\Box({\bf x})}} \, \underbrace{\sum_{k=0}^{n} {\tau^k (1-\tau)^{n-k} \over {(n -k )! }} s_\Box^{n-k}({\bf x}) \sum_{|\mu| = k} s_\mu({\bf x}) \dim(\mu,\lambda)}_{\text{call this s_\lambda({\bf x};\tau)}}$$$$

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.

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

thanks, ines.

• I assume $p_{|v|-|u|}$ should be $p_{|v|}$, right? May 6, 2022 at 20:11
• It's either that or else I should have started at zero, i.e. $p_0 \lhd \cdots \lhd p_{|v|-|u|}$ where $p_0 = u$ and $p_{|v|-|u|}= v$. Not sure which is better. May 6, 2022 at 22:43
• Now it's fixed. May 6, 2022 at 23:00

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.