Let $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ be an infinite sequence of positive real numbers such that following $\Bbb{N} \times \Bbb{N}$ tridiagonal matrix

\begin{equation} T(\Bbb{y}) := \, \begin{pmatrix} 1 & y_1 & 0 & 0 & \\ 1 & 1 & y_2 & 0 & \\ 0 & 1 & 1 & y_3 & \\ 0 & 0 & 1 & 1 & \\ & & & & \ddots \end{pmatrix} \end{equation}

is totally positive in the sense that all of its leading principal minors $[1, \dots,n]$ for $n \geq 1$ are positive; here $[j_1, \dots, j_n]$ denotes the principal minor of $T$ whose row and columns sets are indexed by an (ordered) subset $\{j_1 < \dots < j_n\} \subset \Bbb{N}$. The terminology totally positive is justified since the positivity of $[1, \dots,n]$ for all $n \geq 1$ is in fact equivalent to the positivity of all principal minors of the form $[m,\dots,n]$ with $m \leq n$. I refer the reader to the article of Shih-Wei Yang and Andrei Zelevinsky "Cluster algebras of finite type via coxeter elements and principal minors" as a reference/guide.

For the purposes of this post we'll say that a sequence $\Bbb{y}$ of postive real numbers is totally positive if the corresponding tridiagonal matrix $T(\Bbb{y})$ is totally positive. I should point out that in the current literature a totally positive sequence is usually defined in terms of the total positivity of the corresponding infinite Toeplitz matrix; see for example the work of Lam and Pylyavskyy. One well known method to obtain a totally positive sequence $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ is to choose an auxiliary sequence $\Bbb{t} = \big(t_1, t_2, t_3, \dots \big)$ with $0 < t_n < 1$ for each $n \geq 1$ and then simply set $y_1 = t_1$ and $y_{n+1} = (1 - t_n) \, t_{n+1}$ for all $n \geq 1$. Indeed this recipe gives a parametrization of all totally positive sequences (as I've defined them).

Let us say that a sequence $\Bbb{y}$ is ultra-positive if in addition to being totally positive the quasi-shifted sequence

\begin{equation} \sigma_m(\Bbb{y}) := \, \Big( {y_{m+1} \over {y_m}}, \, y_{m+2}, \, y_{m+3}, \, y_{m+4}, \, \dots \Big) \end{equation}

is totally positive for all $m \geq 1$; equivalently if the infinite tridiagonal matrix

\begin{equation} \displaystyle S_m(\Bbb{y}) := \, \begin{pmatrix} y_m & y_{m+1} & 0 & 0 & \\ 1 & 1 & y_{m+2} & 0 & \\ 0 & 1 & 1 & y_{m+3} & \\ 0 & 0 & 1 & 1 & \\ & & & & \ddots \end{pmatrix} \end{equation}

is totally positive for each $m \geq 1$. In other words the leading principal minors of $S_m(\Bbb{y})$ must be positive for each $m \geq 1$ or more concretely, after using the Laplace expansion, $y_m > y_{m+1}$ and

\begin{equation} y_m[m+2, \dots, n] - y_{m+1}[m+3, \dots,n] > 0 \end{equation}

whenever $m \geq 1$ and $n \geq m+3$. The last inequality can be rewritten as

\begin{equation} \displaystyle {y_{m+1} \over {y_m}} \, < \, \kappa_n(m) := \, {[m+2,\dots,n] \over {[m+3,\dots,n]}} \end{equation}

whenever $m \geq 1$ and $n \geq m+3$. Furthermore, the following short plücker-like relation

\begin{equation} \begin{array}{c} [m, \dots, n] \, [m+1, \dots, n+1] \\ = \\ y_m \cdots y_n \ + \ [m , \dots, n+1] \, [m+1, \dots, n]. \end{array} \end{equation}

together with the fact that $\Bbb{y}$ is totally positive implies that $\kappa_n(m)$ is monotonically decreasing as $n \geq m+3$ increases and therefore

\begin{equation} {y_{m+1} \over {y_m}} \, \leq \, \kappa_\infty(m) := \, \lim_{n \rightarrow \infty} \kappa_n(m) \end{equation}

So we can conclude that $\Bbb{y}$ is ultra-positive if and only if (1) $\Bbb{y}$ is totally positive, (2) the limit $\kappa_\infty(m)$ exists and is positive for all $m \geq 1$, and (3)

\begin{equation} {y_{m+1} \over {y_m}} \, \leq \, \kappa_\infty(m) \end{equation}

for all $m \geq 1$.

An example of an ultra-positive sequence is

\begin{equation} \displaystyle \Bbb{y}_\mathrm{P} := \, \Big({1 \over 2}, \, {1 \over 3}, \, {1 \over 4}, \, \dots \Big) \end{equation}

where $\mathrm{P}$ stands for Plancherel --- the sequence is related to the Plancherel state $\varphi_\mathrm{P}$ of Young-Fibonacci lattice $\Bbb{YF}$ mentioned in this post Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality. In addition any $r$-shift $(y_r, y_{r+1}, y_{r+2}, \dots)$ of an ultra-positive sequence $(y_1, y_2, y_3, \dots)$ is clearly ultra-positive.

Question: How can one produce other examples of ultra-positive sequences and, more generally, how can the space of ultra-positive sequences be parameterized ? More specifically: Is there a variant of the Thoma simplex, perhaps

\begin{equation} \Omega = \Big\{ \underline{\omega} = \big(\omega_1 \geq \omega _2 \geq \omega_ 3 \geq \cdots \big) \, \Big| \, 1 \geq \omega_1 + \omega_2 + \omega_3 + \cdots \, \Big\} \end{equation}

together with a map $\underline{\omega} \mapsto \Bbb{y}_{\underline{\omega}}$ to the space of ultra-positive sequences which is a homeomorphism (in some appropriate topology) ?

thanks, ines.

  • 1
    $\begingroup$ I'm sure you're striving to optimize this post, but please be aware that each edit bumps this post to the top of the stack and other posts (also vying for attention) down. So if you could condense several improvements into an edit, rather than changing by a single character, it would be appreciated. $\endgroup$ – Todd Trimble Nov 21 at 2:12

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