# Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality

$$\newcommand\rank{\lvert#1\rvert}$$Let $$\Bbb{P}$$ be a 1-differential poset with a unique bottom element $$\emptyset \in \Bbb{P}$$. With some minor abuse in terminology, The Plancherel measure state $$\varphi_\mathrm{P}$$ is the real-valued function defined for each $$x \in \Bbb{P}$$ by $$\begin{equation} \varphi_\mathrm{P}(x) \mathrel{:=} \, {1 \over {\, \rank x!} } \dim \bigl(\emptyset, x \bigr), \end{equation}$$ where $$\rank x= n$$ is the rank of the element, $$\dim\bigl(\emptyset, x \bigr)$$ is the number of saturated chains $$x_0 \lhd \, \dotsb \lhd \, x_n$$ in $$\Bbb{P}$$ starting at $$x_0 = \emptyset$$ and ending at $$x_n = x$$, and $$x \lhd \, y$$ signifies the covering relation in the poset. The Plancherel measure is related to $$\varphi_\mathrm{P}$$ as the former can be expressed as the mapping $$\Bbb{P} \ni x \mapsto \varphi_\mathrm{P}(x) \cdot \dim \bigl(\emptyset, x \bigr)$$. Clearly $$\varphi_\mathrm{P}$$ is non-negative (indeed, it's strictly positive), it is normalised (i.e. $$\varphi_\mathrm{P}(\emptyset) = 1$$), and it is harmonic in the sense that $$\begin{equation} \varphi_\mathrm{P}(x) = \, \sum_{x \lhd \, y} \, \varphi_\mathrm{P}(y) \end{equation}$$

Question: Is the Plancherel state $$\varphi_\mathrm{P}$$ always minimal in the sense that whenever $$\varphi_\mathrm{P} = s\phi_1 + (1-s)\phi_2$$ for some parameter $$0 < s < 1$$ and for some pair of non-negative, normalised, harmonic functions $$\phi_1, \phi_2: \Bbb{P} \longrightarrow \Bbb{R}$$ then $$\varphi_\mathrm{P} = \phi_1 = \phi_2$$? If the assertion is not true in general, why is it true, for example, in the case of the Young–Fibonacci Lattice $$\Bbb{P} = \Bbb{YF}$$ ?

regards, ines.

P.S. This question is a follow-up to Minimal harmonic functions on a poset, which was recently posted on Math StackExchange.

• You sure you don't mean $\mathrm{dim}(\varnothing, x)^2$? Nov 13 '20 at 0:04
• Strictly speak the Plancherel measure is $x \mapsto {1 \over {\, |x|!}} \dim^2(\emptyset, x)$ --- that's why I wrote "minor abuse of terminology". Nevertheless I want to consider the mapping $x \mapsto {1 \over {\, |x|!}} \dim(\emptyset, x)$ instead. Nov 13 '20 at 0:07
• Gotcha, thanks. But then I'm not even sure that what you've written is an extrema of the boundary of Young's lattice. Does the Vershik-Kerov stuff still apply? Nov 13 '20 at 0:10
• Goodman and Kerov (for example) indicate in their paper that $\varphi_\mathrm{P}$ is an extrema of the Martin Boundary in the case of the Young lattice $\Bbb{Y}$, although this is not their result. It's unclear whether or not the $\varphi_\mathrm{P}$ is an extrema in the Young-Fibonacci case; which is what I'm most curious about. Nov 13 '20 at 0:22
• The extremality for the case of $\Bbb{YF}$ is proven in "The Plancherel measure of the Young-Fibonacci graph" by Gnedin and Kerov, but it seems to exploit the very specific structure of this poset, so it is unclear how much of it can be generalized to other differential posets. Nov 13 '20 at 0:34