Here's an example of $\Bbb{R}[[q]]$-total positivity arising from specializing an instance of Schur-positivity related to the Grassmannian:

Given an ordered $k$-subset $I = \{i_1 < \dots < i_k \}$ taken from $\{1, \dots, n\}$ let $s_I$ denote the Schur function associated to the partition $\big(i_k - k, \, \dots, \, i_1 - 1 \big)$ and let $[I]$ denote the corresponding Plücker coordinate on Grassmannian $\mathrm{Gr}_{k,n}$. The mapping $[I] \stackrel{s}{\mapsto} s_I$ extends to a ring homomorphsim from the homogeneous coordinate ring $\Bbb{C}\Big[\widehat{\mathrm{Gr}}_{k,n}\Big]$ of the (affine cone over of the) Grassmannian $\mathrm{Gr}_{k,n}$ to the ring of symmetric functions. Now take the principal specialization $x_i = q^{i-1}$ for $i\geq 1$ of the Schur functions. In this way we obtain a point $\mathrm{P}(q)$ in $\widehat{\mathrm{Gr}}_{k,n}$ with the property that

\begin{equation} \begin{array}{ll} \displaystyle [I](\mathrm{P}(q)) &\displaystyle = \, s_I \big(1,q,q^2, \dots \big) \\ \\ &\displaystyle = \, q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle\mathrm{boxes}}{b \in \lambda}} \, \Big( 1 - q^{ \, \mathrm{hook}(b)} \Big)^{-1} \end{array} \end{equation}

which is clearly $\Bbb{R}[[q]]$-totally positive upon expansion.

regards, ines.

p.s. I say an instance of Schur-positivity because, conjecturally, $s(\phi)$ will be Schur-positive for any cluster variable $\phi$ generated inside $\Bbb{C}\Big[ \widehat{\mathrm{Gr}}_{k,n}\Big]$. If the conjecture is true then $\phi(\mathrm{P}(q))$ will also be $\Bbb{R}[[q]]$-positive.

Here's an example of $\Bbb{R}[[q]]$-total positivity arising from specializing an instance of Schur-positivity related to the Grassmannian:

Given an ordered $k$-subset $I = \{i_1 < \dots < i_k \}$ taken from $\{1, \dots, n\}$ let $s_I$ denote the Schur function associated to the partition $\big(i_k - k, \, \dots, \, i_1 - 1 \big)$ and let $[I]$ denote the corresponding Plücker coordinate on Grassmannian $\mathrm{Gr}_{k,n}$. The mapping $[I] \stackrel{s}{\mapsto} s_I$ extends to a ring homomorphsim from the homogeneous coordinate ring $\Bbb{C}\Big[\widehat{\mathrm{Gr}}_{k,n}\Big]$ of the (affine cone over of the) Grassmannian $\mathrm{Gr}_{k,n}$ to the ring of symmetric functions. Now take the principal specialization $x_i = q^{i-1}$ for $i\geq 1$ of the Schur functions. In this way we obtain a point $\mathrm{P}(q)$ in $\widehat{\mathrm{Gr}}_{k,n}$ with the property that

\begin{equation} \begin{array}{ll} \displaystyle [I](\mathrm{P}(q)) &\displaystyle = \, s_I \big(1,q,q^2, \dots \big) \\ \\ &\displaystyle = \, q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle\mathrm{boxes}}{b \in \lambda}} \, \Big( 1 - q^{ \, \mathrm{hook}(b)} \Big)^{-1} \end{array} \end{equation}

the coefficients of which are clearly positive integers upon expansion in $\Bbb{R}[[q]]$. One might say the point $\mathrm{P}(q)$ is a $\Bbb{R}[[q]]$-totally positive point of the Grassmannian.

regards, ines.

p.s. I say an instance of Schur-positivity because, conjecturally, $s(\phi)$ will be Schur-positive for any cluster variable $\phi$ generated inside $\Bbb{C}\Big[ \widehat{\mathrm{Gr}}_{k,n}\Big]$. If the conjecture is true then $\phi(\mathrm{P}(q))$ will also be $\Bbb{R}[[q]]$-positive.

Here's an example of $\Bbb{R}[[q]]$-total positivity arising from specializing an instance of Schur-positivity related to the Grassmannian:

Given an ordered $k$-subset $I = \{i_1 < \dots < i_k \}$ taken from $\{1, \dots, n\}$ let $s_I$ denote the Schur function associated to the partition $\big(i_k - k, \, \dots, \, i_1 - 1 \big)$ and let $[I]$ denote the corresponding Plücker coordinate on Grassmannian $\mathrm{Gr}(k,n)$. The mapping $[I] \mapsto s_I$ extends to a ring homomorphsim from the homogeneous coordinate ring $\Bbb{C}\Big[\widehat{\mathrm{Gr}}(k,n) \Big]$ of the (affine cone over of the) Grassmannian $\mathrm{Gr}(k,n)$ to the ring of symmetric functions. Now take the principal specialization $x_i = q^{i-1}$ for $i\geq 1$ of the Schur functions. In this way we obtain a point $\mathrm{P}(q)$ in $\widehat{\mathrm{Gr}}(k,n)$ with the property that

\begin{equation} \begin{array}{ll} \displaystyle [I](\mathrm{P}(q)) &\displaystyle = \, s_I \big(1,q,q^2, \dots \big) \\ \\ &\displaystyle = \, q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle\mathrm{boxes}}{b \in \lambda}} \, \Big( 1 - q^{ \, \mathrm{hook}(b)} \Big)^{-1} \end{array} \end{equation}

which is clearly $\Bbb{R}[[q]]$-totally positive upon expansion.

regards, ines.

Here's an example of $\Bbb{R}[[q]]$-total positivity arising from specializing an instance of Schur-positivity related to the Grassmannian:

Given an ordered $k$-subset $I = \{i_1 < \dots < i_k \}$ taken from $\{1, \dots, n\}$ let $s_I$ denote the Schur function associated to the partition $\big(i_k - k, \, \dots, \, i_1 - 1 \big)$ and let $[I]$ denote the corresponding Plücker coordinate on Grassmannian $\mathrm{Gr}_{k,n}$. The mapping $[I] \stackrel{s}{\mapsto} s_I$ extends to a ring homomorphsim from the homogeneous coordinate ring $\Bbb{C}\Big[\widehat{\mathrm{Gr}}_{k,n}\Big]$ of the (affine cone over of the) Grassmannian $\mathrm{Gr}_{k,n}$ to the ring of symmetric functions. Now take the principal specialization $x_i = q^{i-1}$ for $i\geq 1$ of the Schur functions. In this way we obtain a point $\mathrm{P}(q)$ in $\widehat{\mathrm{Gr}}_{k,n}$ with the property that

\begin{equation} \begin{array}{ll} \displaystyle [I](\mathrm{P}(q)) &\displaystyle = \, s_I \big(1,q,q^2, \dots \big) \\ \\ &\displaystyle = \, q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle\mathrm{boxes}}{b \in \lambda}} \, \Big( 1 - q^{ \, \mathrm{hook}(b)} \Big)^{-1} \end{array} \end{equation}

which is clearly $\Bbb{R}[[q]]$-totally positive upon expansion.

regards, ines.

p.s. I say an instance of Schur-positivity because, conjecturally, $s(\phi)$ will be Schur-positive for any cluster variable $\phi$ generated inside $\Bbb{C}\Big[ \widehat{\mathrm{Gr}}_{k,n}\Big]$. If the conjecture is true then $\phi(\mathrm{P}(q))$ will also be $\Bbb{R}[[q]]$-positive.