This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $Bbb{YF}$, namely:
Differential posets, the Plancherel state $varphi_mathrm{P}$, and minimality
Young-Fibonacci tableaux, content, and the Okada algebra
Infinite tridiagonal matrices and a special class of totally positive sequences
According to the results of Goodman and Kerov, the parameter space $Omega$ (as a set) of the Martin Boundary $E$ of the Young-Fibonacci lattice $Bbb{YF}$
consists of: (1) an outlier point $mathrm{P}$ together with (2) all pairs $(w,beta)$ where $0 < beta leq 1$
is a real parameter and $w = cdots a_4 a_3 a_2 a_1$ is an infinite fibonacci word with
$2$‘s occurring at positions $ dots, d_4, d_3, d_2, d_1$ when read from right to left
such that $sum_{i geq 1} 1/d_i < infty$. Note that the position
of $2$ in a fibonacci word of the form $w = u2v$ is $1 + |v|$ where $|v|$
denotes the sum of the digits of the suffix $v$, otherwise called
the length of $v$. The reader should consult Goodman and Kerov’s paper for a description of $Omega$‘s topology.
To each $omega in Omega$ the corresponding point $varphi_omega in E$,
is a non-negative, normalised harmonic function on $Bbb{YF}$. Under this correspondence $varphi_mathrm{P}$
is the Plancherel state, i.e. for $u in Bbb{YF}$
begin{equation}
label{plancherel-measure}
varphi_mathrm{P}(u) := {1 over {, |u|!}} , mathrm{dim}big( emptyset, u big)
end{equation}
where $dim(u,v)$ denotes the number of saturated chains $(u_0 lhd cdots lhd , u_n)$ in $Bbb{YF}$ starting at $u_0 = u$ and ending at $u_n =v$.
Recently Vsevolod Evtushevsky (see arXiv:2012.07447 and arXiv:2012.08107)
has announced a proof showing that the Martin Boundary $E$ coincides with its minimal boundary. If this is true, then any positive, normalised harmonic function
$varphi: Bbb{YF} longrightarrow Bbb{R}$ should be expressed as
begin{equation}
varphi(u) = int_Omega dM_varphi(omega) , varphi_omega(u)
end{equation}
where $dM_varphi$ is
a measure (morally a boundary condition) uniquely determined by $varphi$.
There is an alternative supply of normalised harmonic functions on the Young-Fibonacci lattice: I’ll call them Okada-Schur functions,
but strictly speaking they are commutative versions of the polynomials defined in two non-commutative variables as introduced by Okada (Goodman and Kerov call them clone symmetric functions):
Let ${bf y} = (y_1, y_2, y_3, dots)$ be a sequence of real numbers.
The Okada-Schur function $sigma_{bf y}: Bbb{YF} longrightarrow Bbb{R}$
associated to the sequence ${bf y}$ is defined recursively (with respect to length) by
begin{equation}
sigma_{bf y}(u) =
left{
begin{array}{ll}
T_k ({bf y})
& text{if $u = 1^k$ for some $k geq 0$} \ \
S_k big({{bf y} + |v|} big) cdot sigma_{bf y}(v)
& text{if $u=1^k2v$ for some $k geq 0$}
end{array}
right.
end{equation}
where
begin{equation}
T_ell ({bf y}) =
det
underbrace{begin{pmatrix}
1 & y_1 & 0 & cdots\
1 & 1 & y_2 &\
0 & 1 & 1 & \
vdots & & & ddots
end{pmatrix}}_{text{$ell times ell $ tridiagonal matrix}}
quad
S_{ell -1} ({bf y}) =
det
underbrace{begin{pmatrix}
y_1 & y_2 & 0 & cdots\
1 & 1 & y_3 &\
0 & 1 & 1 & \
vdots & & & ddots
end{pmatrix}}_{text{$ell times ell$ tridiagonal matrix}}
end{equation}
for integers $ell geq 1$ and
where we employ the notation ${bf y} + r:= (y_{1+r}, , y_{2+r}, , y_{3+r}, , dots)$ for any integer $r geq 0$.
Okada proved that $sigma_{bf y}$ is a normalised harmonic function for any infinite sequence ${bf y}$. It is not
clear to me what are necessary and sufficient conditions for $sigma_{bf y}$ to be positive, let alone non-negative for that matter.
There is, however, at least one case when $sigma_{bf y}$ is
positive, namely:
Remark 1:
Let ${bf y} = big({1 over 2}, {1 over 3}, {1 over 4}, dots big)$
then $sigma_{bf y} = varphi_mathrm{P}$ (i.e. the Plancherel state).
Furthermore
Remark 2: If ${bf y} = (y_1, y_2, y_3, dots)$ is any sequence of real numbers for which $sigma_{bf y}$ is positive
then $sigma_{{bf y} + r}$ will be positive for any integer $r geq 0$. (This basically follows from some of the observations made in Infinite tridiagonal matrices and a special class of totally positive sequences).
Question 1: Suppose ${bf y}$ is a sequence such that $sigma_{bf y}$
is positive. Under which circumstances will $sigma_{bf y} in Omega$ ?
My guess is only when ${bf y} = big({1 over 2}, {1 over 3}, {1 over 4}, dots big)$ but what about ${bf y} =
big({1 over {r+2}}, {1 over {r+3}}, {1 over {r+4}}, dots big)$ for $r geq 1$ ?
Question 2: Suppose instead that $sigma_{bf y}$ is positive but $sigma_{bf y}
notin Omega$. What is the unique measure $dM_{bf y}$ on the Martin boundary $Omega$ for which
begin{equation}
sigma_{bf y}(u) = int_Omega dM_{bf y}(omega) , varphi_omega(u)
end{equation}
Again, what about ${bf y} = big({1 over {r+2}}, {1 over {r+3}}, {1 over {r+4}}, dots big)$ for $r geq 1$ ?
thanks, ines.
