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\topmatter
\affil
Department of Mathematics\\
The University of Pennsylvania\\
Philadelphia, PA 19104-6395, U.S.A. \\
{\tt borodine\@math.upenn.edu}\\
\\
Dobrushin Mathematics Laboratory,\\ Institute for
Problems of Information Transmission,\\ Bolshoy Karetny 19, 101447
Moscow GSP-4, RUSSIA. \\
{\tt olsh\@iitp.ru, olsh\@glasnet.ru}
\endaffil
\date
Submitted: November 22, 1999; Accepted: May 15, 2000.
\enddate
\subjclass Primary 05E10, Secondary 31C20, 60C05\endsubjclass
\thanks
One of the authors (G.O.) was supported by the Russian Foundation for Basic Research under grant 98--01--00303.
\endthanks
\title Harmonic functions on multiplicative graphs and interpolation
polynomials
\endtitle
\author Alexei Borodin and Grigori Olshanski
\endauthor
\abstract We construct examples of nonnegative harmonic functions on
certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on
the infinite symmetric group. Our method relies on multivariate
interpolation polynomials associated with Schur's S and P functions
and with Jack symmetric functions. As a by--product, we compute
certain Selberg--type integrals.
\endabstract
\toc
\widestnumber\head{\S7.}
\head \S0. Introduction \endhead
\head \S1. The general formalism\endhead
\head \S2. The Young graph \endhead
\head \S3. The Jack graph \endhead
\head \S4. The Kingman graph \endhead
\head \S5. The Schur graph \endhead
\head \S6. Finite--dimensional specializations \endhead
\subhead 6.1. Truncated Young branching \endsubhead
\subhead 6.2. $\Gamma$--shaped Young branching \endsubhead
\subhead 6.3. Truncated Kingman branching \endsubhead
\subhead 6.4. Truncated Schur branching \endsubhead
\head \S7. Appendix \endhead
\head{} References \endhead
\endtoc
\endtopmatter
\font\smcp=cmcsc8
\headline={\ifnum\pageno>1 {\smcp the electronic journal of combinatorics {\bf 7} (2000), \#R28\hfill\folio} \fi}
\document
\head \S0. Introduction \endhead
Let $\Y$ denote the lattice of Young diagrams ordered by inclusion.
For $\mu,\la\in\Y$, we write $\la\searrow\mu$ if $\la$ covers $\mu$,
i.e., $\la$ differs from $\mu$ by adding a box. We consider $\Y$ as a
graph whose vertices are arbitrary Young diagrams $\mu$ and the edges
are couples $(\mu,\la)$ such that $\la\searrow\mu$. We shall call $\Y$ the {\it Young graph}.
A function $\varphi(\mu)$ is called a {\it harmonic function} on the Young
graph \cite{VK} if it satisfies the condition
$$
\varphi(\mu)=\sum_{\la:\, \la\searrow \mu} \varphi(\la), \qquad
\forall\mu\in\Y. \tag0.1
$$
We are interested in nonnegative harmonic functions $\varphi$
normalized at the empty diagram: $\varphi(\varnothing)=1$. Such
functions form a convex set denoted as $\Harm^+_1(\Y)$.
The functions $\varphi\in\Harm^+_1(\Y)$ have an important
representation--theoretic meaning: they are in a natural bijective
correspondence with central, positive definite, normalized functions
on the infinite symmetric group $S(\infty)$, see \cite{VK},
\cite{KV2}. Thoma's description of characters on $S(\infty)$ means
that the extreme points of $\Harm^+_1(\Y)$ form an
infinite--dimensional simplex $\Om$ (called the Thoma simplex), see
\cite{T}, \cite{VK}, \cite{KV2}, \cite{W}. For general elements
$\varphi\in\Harm^+_1(\Y)$, there is a (unique) Poisson--type
integral representation,
$$
\varphi(\la)=\int_\Om K(\la,\om)\, P(d\om), \qquad \forall \la\in\Y,
\tag0.2
$$
where $P$ is a probability measure on $\Om$ (the `boundary measure'
for $\varphi$) and $K(\la,\om)$ is a positive function on
$\Y\times\Om$ (the `Poisson kernel' or `Martin kernel' for $\Y$), see
\cite{KOO}. Note that any probability measure $P$ on $\Om$ gives rise
to an element of $\Harm^+_1(\Y)$; in particular, the extreme
$\varphi$'s are exactly the functions $K(\cdot,\om)$ corresponding to
Dirac measures on $\Om$.
This abstract result shows how large $\Harm^+_1(\Y)$ is but it does
not explain how to construct explicitly nonextreme
functions $\varphi$ or what nonextreme $\varphi$'s could be
interesting for applications.
Concrete examples of nonextreme functions $\varphi$ first emerged in
\cite{KOV} in connection with a problem of harmonic analysis on the
infinite symmetric
group $S(\infty)$. These functions, denoted as $\varphi_{zz'}$,
depend on two parameters, and the corresponding `boundary measures'
$P_{zz'}$ govern the spectral decomposition of certain natural unitary
representations. \footnote{The measures $P_{zz'}$ are very
interesting objects. They are studied in detail in our papers
\cite{P.I} -- \cite{P.V}, \cite{BO1}, \cite{BO2}.}
The explicit expression of the functions $\varphi_{zz'}$ (see formula
\tht{2.4} below) has an interesting combinatorial structure which
raises a number of questions. For instance, one can ask whether
there exist similar families of harmonic functions for other graphs.
The answer is affirmative: \cite{B1}, \cite{Ke5}. \footnote{Another
question, a characterization of the functions of type
$\varphi_{zz'}$, was examined in \cite{B1}, \cite{Ro}.}
The paper \cite{B1} concerns the graph $\SS$ of shifted Young
diagrams which is related to projective representations of the
symmetric groups.
The paper \cite{Ke5} contains a generalization in another direction:
a deformation of the family $\{\varphi_{zz'}\}$, which is consistent
with a deformation of the basic equation \tht{0.1}:
$$
\varphi(\mu)=\sum_{\la:\, \la\searrow\mu}\dimth(\mu,\la)\varphi(\la),
\qquad \forall \mu. \tag0.3
$$
Here $\th>0$ is the deformation parameter and $\dimth(\mu,\la)>0$ are the
coefficients that arise in (the simplest case of) Pieri's rule
for Jack symmetric functions with parameter $\th$. The initial
situation corresponds to the particular value $\th=1$, when Jack
symmetric functions coincide with Schur's $S$--functions.
Note that in the limit as $\th\to0$, the harmonicity condition
\tht{0.3} essentially coincides with the relation which defines
partition structures in the sense of Kingman \cite{Ki1}, \cite{Ki2},
while the two-parameter family of harmonic functions constructed in
\cite{Ke5} degenerates to the famous Ewens partition structures
\cite{Ew} and its generalization due to Pitman, see
\cite{Pi}, \cite{PY}, \cite{Ke4}.
In the present paper, we propose a simple combinatorial construction,
which allows us to get, in a unified way, all these concrete examples of
harmonic functions as well as some new ones. In the new examples, the
`boundary measures' $P$ are supported by finite--dimensional simplices, and the Poisson integral representation leads to certain
Selberg--type integrals. \footnote{A connection between Poisson
integral representation of type \tht{0.2} and Selberg integrals was
first exploited in \cite{Ke3}.}
Our construction relies on the so--called shifted (or factorial)
versions of Schur's $S$ and $P$ functions and of Jack symmetric
functions. These new combinatorial functions arise in different
topics, see, e.g., \cite{S}, \cite{KS}, \cite{OO}, \cite{OO2}, \cite{Ok1}, \cite{Ok2}. They are also called interpolation
polynomials, because they give solutions to certain multivariate
interpolation problems.
The paper is organized as follows. In \S1, we expose the general
formalism. In \S2, it is applied to the Young graph to derive the
family $\{\varphi_{zz'}\}$. In \S\S3--5, we apply it to the Young
graph with Jack edge multiplicities $\J$, next to the Kingman graph
$\K$, and then to the Schur graph $\SS$; the arguments are quite
similar. Section 6 is devoted to constructing harmonic functions
of a different sort --- those with finite--dimensional `boundary
measures'; here we also evaluate Selberg--type integrals. The
final \S7 is an appendix on the Poisson integral representation.
\head \S1. The general formalism \endhead
In this section, we deal with an abstract graph $\G$ satisfying
certain conditions listed below. In the next sections, concrete
examples of $\G$ will be considered.
Our assumptions and conventions concerning $\G$ are as follows:
$\bullet$ To simplify the notation, we identify the graph with its
set of vertices.
$\bullet$ The vertices are partitioned into levels,
$\G=\G_0\sqcup\G_1\sqcup\G_2\sqcup\dots$, so that the endpoints of
any edge lie on consecutive levels. That is, $\G$ is a graded graph.
$\bullet$ The level of a vertex $\mu$ is denoted as $|\mu|$. If two
vertices $\mu,\la$ form an edge, $|\la|=|\mu|+1$, then we write
$\la\searrow\mu$ or $\mu\nearrow\la$.
$\bullet$ All the levels $\G_n$ are finite.
$\bullet$ The lowest level $\G_0$ consists of a single
vertex denoted as $\varnothing$.
$\bullet$ For any vertex $\mu$ there exists at least one vertex
$\la\searrow\mu$ and for any vertex $\la\ne\varnothing$ there exists
at least one vertex $\mu\nearrow\la$. This implies that the graph is
connected.
(Our main example is the Young graph, see \S2. )
$\bullet$ Finally, assume that we are given an {\it edge multiplicity
function\/} which assigns to any edge $\mu\nearrow\la$ a strictly
positive number $\ka(\mu,\la)$ --- its formal multiplicity. It should
be emphasized that these numbers are not necessarily integers.
(For the Young graph, all the formal multiplicities are equal to 1;
graphs with nontrivial multiplicities are considered in \S3 and \S4.)
A complex function $\varphi(\mu)$ on $\G$ is called a {\it harmonic function
on the graph\/} $\G$ if it satisfies the relation
$$
\varphi(\mu)=\sum_{\la:\, \la\searrow\mu} \ka(\mu,\la)\varphi(\la)
\tag 1.1
$$
for any vertex $\mu$ (the sum in the right--hand side is finite, because all the
levels are finite). Let $\Harm(\G)$ denote the space of all harmonic
functions endowed with the topology of pointwise convergence. Let
$\Harm^+(\G)$ be the subset of nonnegative harmonic functions and
$\Harm^+_1(\G)$ be the subset of the functions
$\varphi\in\Harm^+(\G)$ with the normalization
$\varphi(\varnothing)=1$.
Clearly, $\Harm^+_1(\G)$ is a convex subset of $\Harm(\G)$. Moreover,
it is a compact measurable space (here we again employ the
finiteness assumption). We shall use some well--known general
theorems about convex compact measurable sets which can be found,
e.g., in \cite{Ph}.
Let $\Om(\G)$ denote the set of extreme points in $\Harm^+_1(\G)$.
This is a set of type $G_\delta$, hence, a Borel measurable set. Given
$\om\in\Om(\G)$, let us denote by $K(\,\cdot\,, \om)$ the
corresponding extreme harmonic function on $\G$. Note that
$K(\mu,\,\cdot\,)$ is a Borel measurable function on $\Om(\G)$ for
any fixed $\mu\in\G$.
\proclaim{Theorem 1.1} For each element $\varphi\in\Harm^+_1(\G)$
there exists a unique probability measure $P$ on $\Omega(\G)$ such
that
$$
\varphi(\mu)=\int_{\Om(\G)} K(\mu,\om) P(d\om), \qquad
\forall\mu\in\G. \tag 1.2
$$
\endproclaim
\demo{Proof} See \S7. \qed
\enddemo
We call \tht{1.2} the {\it Poisson integral representation\/} of the
function $\varphi$.
To any {\it path} $\tau$ going from a vertex $\mu$ to a vertex $\la$ with
$|\la|>|\mu|$,
$$
\tau=(\mu=\la_0\nearrow\la_1\nearrow\dots\nearrow\la_k=\la),
\qquad k=|\la|-|\mu|,
$$
we assign its weight
$$
w(\tau)=\prod_{i=1}^k\ka(\la_{i-1},\la_i)
$$
and then set
$$
\dimG(\mu,\la)=\sum_\tau w(\tau),
\tag 1.3
$$
summed over all paths from $\mu$ to $\la$. We extend this definition
to all couples $(\mu,\la)$ by agreeing that $\dimG(\mu,\mu)=1$ and
$\dim(\mu,\la)=0$ if $\mu\ne\la$ are such that there is no path from
$\mu$ to $\la$. Next, we set $\dimG\la=\dimG(\varnothing,\la)$.
In all examples of the graphs $\G$ considered in the present paper
one can embed (the vertices of) $\G$ into $\Om(\G)$ in such a way that
any point $\om\in\Om(\G)$ can be approximated by a sequence of vertices
$\{\la(n)\in\G_n\}_{n=1,2,\dots}$, and for any such sequence
$$
K(\mu,\om)=\lim_{n\to\infty}\frac{\dimG(\mu,\la(n))}{\dimG\la(n)}\,.
$$
Given a function $\varphi\in\Harm^+_1(\G)$, we set for each $n$
$$
M_n(\la)=\dimG\la\cdot\varphi(\la), \qquad \la\in\G_n.
\tag 1.4
$$
Using the harmonicity relation \tht{1.1} and induction on $n$ one
readily verifies that $\sum_{\la\in\G_n} M_n(\la)=1$. Thus, each $M_n$ is a
probability distribution on $\G_n$.
For all examples of the graphs $\G$ considered in this paper one can
transfer the measure $M_n$ to $\Om(\G)$ via the
embedding $\G\hookrightarrow\Om(\G)$ mentioned above. Then the
measure $P$ appearing in the integral representation \tht{1.2} is the
weak limit of the measures $M_n$ as $n\to\infty$.
We say that $(\G,\ka(\,\cdot\,,\,\cdot\,))$ is a {\it multiplicative
graph\/} \cite{KV1}, \cite{KV2}, if the following conditions are satisfied. First,
the 1st floor $\G_1$ consists of a single vertex denoted by the
symbol ``$(1)$''.
Next, there exists a graded commutative unital algebra $A$ over $\R$,
$A=A_0+A_1+\dots$, and a homogeneous basis $\{\P_\mu\}$ in $A$ indexed
by the vertices $\mu\in\G$, such that $\P_\varnothing=1$ and $\deg
\P_\mu=|\mu|$. Finally, for any $\mu$,
$$
\P_\mu \P_{(1)}=\sum_{\la:\,\la\searrow\mu} \ka(\mu,\la)\P_\la\,.
\tag 1.5
$$
Note that this implies $\ka(\varnothing, (1))=1$. (All the graphs
considered in the present paper are multiplicative. For instance, in
the case of the Young graph, the algebra $A$ is the algebra of
symmetric functions and the basis $\{\P_\mu\}$ is formed by the Schur
functions.)
Iterating the
relation \tht{1.5} we get the expansion
$$
\P_{(1)}^n=\sum_{\la\in\G_n}\dimG\la\cdot\P_\la\,,
\tag 1.6
$$
which is a useful tool for computing the dimensions $\dimG\la$. More
generally, given $\mu\in\G_m$ and $n>m$,
$$
\P_\mu\P_{(1)}^{n-m}=\sum_{\la\in\G_n}\dimG(\mu,\la)\cdot\P_\la\,.
\tag 1.7
$$
\proclaim{Theorem 1.2 \cite{KV1}} Let $\G$ be a
multiplicative graph and let $A$ be the
corresponding algebra. Given $\varphi\in\Harm^+_1(\G)$, let
$\pi:A\to\C$ be the linear functional sending each $\P_\mu$ to
$\varphi(\mu)$.
Then $\varphi$ is extreme if and only if $\pi$ is multiplicative.
\endproclaim
Note that a linear functional $\pi:A\to\C$ corresponds to a function
$\varphi\in\Harm^+_1(\G)$ if and only if $\pi(1)=1$, $\pi(\P_\mu)\ge0$
for any $\mu$, and $\pi$ factors through $A/(\P_{(1)}-1)A$.
Now we shall explain our method of producing harmonic functions.
Assume $A^*$ is a commutative algebra\footnote{The superscript $\ast$ does not
mean the passage to a dual space.}, $\{\P^*_\mu\}$ is a family of elements in
$A^*$ indexed by the vertices $\mu\in\G$, $\P^*_\varnothing=1$. We
assume that these data obey the following
condition which is a generalization of \tht{1.5}:
$$
\P^*_\mu \P^*_{(1)}=a_n \P^*_\mu+
\sum_{\la:\,\la\searrow\mu} \ka(\mu,\la)\P^*_\la\,, \qquad n=|\mu|,
\tag 1.8
$$
for any $\mu$, where $a_0=0,a_1,a_2,\dots$ is a sequence of numbers.
\proclaim{Proposition 1.3} Under the above assumptions, let $\pi:A^*\to\C$
be a multiplicative linear functional, and let
$$
s=\pi(\P^*_{(1)}), \quad t=-s=-\pi(\P^*_{(1)}).
\tag 1.9
$$
Assume that
$$
s\ne0,a_1,a_2,\dots\,, \quad \text{i.e.}, \quad
t\ne0,-a_1,-a_2,\dots\,.
\tag 1.10
$$
Then the function
$$
\varphi(\mu)=\frac{\pi(\P^*_\mu)}{s(s-a_1)\dots(s-a_{n-1})}
=\frac{(-1)^n\pi(\P^*_\mu)}{t(t+a_1)\dots(t+a_{n-1})}\,,
\qquad n=|\mu|,
\tag 1.11
$$
is harmonic on $\G$.
\endproclaim
We agree that the denominator in \tht{1.11} equals 1 for
$\mu=\varnothing$, so that $\varphi(\varnothing)=1$.
\demo{Proof} Applying $\pi$ to the relation \tht{1.8}
we get
$$
\pi(\P^*_\mu)(s-a_n)=\sum_{\la:\, \la\searrow\mu}
\ka(\mu,\la)\pi(\P^*_\la).
$$
Dividing the both sides by $s(s-a_1)\dots(s-a_n)$ (which is possible
thanks to \tht{1.10}) we get exactly the harmonicity relation \tht{1.1} for
$\varphi$. \qed
\enddemo
A trivial example is $A^*=A$, $\P^*_\mu=\P_\mu$, $a_n\equiv0$. Then, by
Theorem 1.2, $\varphi$ is extreme provided that it is nonnegative. As
we aim to construct interesting examples of nonextreme harmonic
functions, we shall deal either with an algebra $A^*$ distinct from
$A$ or, for $A^*=A$, with a family $\{\P^*_\mu\}$ distinct from
$\{\P_\mu\}$.
In all the examples below, $A^*$ is a filtered
algebra such that the associated graded algebra $\operatorname{gr}A^*$ is
canonically isomorphic to $A$. Thus, with any element of $A^*$ of
degree $\le n$ one can associate its {\it highest term\/} which is a
homogeneous element of $A$ of degree $n$. In our examples, the
highest term of $\P^*_\mu$ coincides with $\P_\mu$. Furthermore,
the algebra $A^*$ can be interpreted, in a certain natural way, as an
algebra of functions on the vertices of $\G$. Thus, for any $f\in
A^*$ and $\la\in\G$, the value $f(\la)$ is well--defined. It turns out
that the elements $\P^*_\mu$ can be characterized by the following
\example{Interpolation Property} Given $\mu\in\G$,
$\mu\ne\varnothing$, $\P^*_\mu$ is the only (up to a scalar factor)
element of degree $|\mu|$ such that
$\P^*_\mu(\la)=0$ for any $\la\ne\mu$ with $|\la|\le|\mu|$.
\endexample
The fact that the highest term of an element $\P^*_\mu$ defined in
this way turns out to be proportional to $\P_\mu$ seems to be rather
surprising. We normalize $\P^*_\mu$ in such a way that its highest
term is exactly equal to $\P_\mu$.
Next, it turns out that $\P^*_{(1)}(\mu)=|\mu|$. Then a simple formal
argument shows that \tht{1.8} holds with $a_n=n$ for any $n=0,1,\dots$ .
Moreover,
$$
\frac{\dimG(\mu,\la)}{\dimG\la}=\frac{\P^*_\mu(\la)}{N(N-1)\dots(N-n+1)}\,,
\qquad \mu\in\G_n, \quad \la\in\G_N, \quad n\le N.
\tag 1.12
$$
The argument is due to Okounkov \cite{Ok1}; it is also reproduced in
\cite{OO}.
{}From now on we shall assume that $a_n=n$.
Then the denominator in the right--hand side of \tht{1.11} will be equal
to $(t)_n=t(t+1)\cdots (t+n-1)$, and \tht{1.11} will take the form
$$
\varphi(\mu)=\frac{(-1)^n\,\pi(\P^*_\mu)}{(t)_n}\,,
\qquad t=-\pi(\P^*_{(1)}), \quad n=|\mu|.
\tag 1.13
$$
Similarly, the formula \tht{1.12} can be rewritten as follows
$$
\frac{\dimG(\mu,\la)}{\dimG\la}=\frac{(-1)^n\,
\P^*_\mu(\la)}{(-N)_n}\,,
\qquad \mu\in\G_n, \quad \la\in\G_N, \quad n\le N.
\tag 1.14
$$
Note that, for any fixed $\la$, the left--hand side of \tht{1.14} satisfies
the harmonicity relation \tht{1.1} provided that $nN$ the
denominator in the right--hand side vanishes). On the
other hand, the expression in the right--hand side of \tht{1.14} is a
particular case of that in the right--hand side of \tht{1.13}: here $\pi$
is the evaluation functional $\pi_\la: f\mapsto f(\la)$ and $t=-N$.
This makes it possible to interpret the construction of Proposition
1.3 as follows: we extrapolate the relation \tht{1.14} from the points
$\la\in\G$, which we identify with the corresponding evaluation
functionals $\pi_\la$, to abstract multiplicative functionals.
A function $\varphi\in\Harm^+_1(\G)$ will be called {\it nondegenerate\/}
if $\varphi(\mu)\ne0$ for all $\mu\in\G$; otherwise it will be called {\it
degenerate}.
\head \S2. The Young graph \endhead
The fundamental example of a graded graph $\G$ is the {\it Young graph\/}
$\Y$ \cite{VK}, \cite{KV2}. By definition, the vertices of $\Y$ are the Young
diagrams including the empty diagram $\varnothing$, the $n$-th floor
$\Y_n$ consists of the diagrams with $n$ boxes, and $\mu\nearrow\la$
means that $\la$ is obtained from $\mu$ by adding a single box. The
numbers $\ka(\mu,\la)$ are all equal to 1. In this section the
symbols $\mu,\la$ are used to denote Young diagrams.
The graph $\Y$ is multiplicative in the sense of the definition given
in \S1: here the algebra $A$
is the algebra $\La$ of symmetric functions, the basis elements
$\P_\mu$ are the Schur functions $s_\mu$, and the relation \tht{1.5}
turns into a special case of the Pieri rule for the Schur functions,
$$
s_\mu s_{(1)}=\sum_{\la:\,\la\searrow\mu} s_\la\,,
\tag 2.1
$$
which is equivalent (under the characteristic map, see \cite{M, I.7}) to
the Young branching rule for irreducible characters of symmetric
groups. For the Young graph, the expansion \tht{1.6}
takes the form
$$
s_{(1)}^n=\sum_{\la:\, |\la|=n}
\dim\la\cdot s_\la,
\tag 2.2
$$
where $\dim\la=\dim_\Y\la$ is the number of standard Young tableaux of
shape $\la$.
Let
$b=(i,j)$ be a box of $\mu$; here $i,j$ are the row number and
the column number of $b$. Recall the definition of the {\it
content\/}, the {\it arm--length\/} and the {\it
leg--length\/} of $b$:
$$
c(b)=j-i,\quad
a(b)=\mu_i-j,\quad
l(b)=\mu'_j-i,
\tag 2.3
$$
where $\mu'$ is the transposed diagram.
\proclaim{Theorem 2.1} Let $z,z'$ be arbitrary complex numbers
and $t=zz'$. Assume that $t\ne0,-1,-2,\dots$ . Then the
following expression is a harmonic function on the Young graph:
$$
\varphi_{zz'}(\mu)=\frac1{(t)_n}
\prod_{b\in\mu}\frac{(z+c(b))(z'+c(b))}{a(b)+l(b)+1}\,,
\quad n=|\mu|.
\tag 2.4
$$
The harmonic functions \tht{2.4} fit into the general scheme of
Proposition 1.3 with the algebra $A^*$ and the family $\{\P^*_\mu\}$
as specified below.
\endproclaim
The first claim of the theorem (harmonicity of $\varphi_{zz'}$)
follows from the computation of a spherical function in \cite{KOV}.
Various direct combinatorial proofs for this claim were given by
Kerov, Postnikov, and Borodin. Kerov's approach is explained in
\cite{Ke5}; actually, in that paper a more general result is
obtained, see Theorem 3.1 below. Postnikov's argument was not
published. Borodin's argument is, perhaps, the most direct and
elementary; it was given in the appendix to \cite{P.I};
actually, the present paper originated from our discussion of that
argument.
For the proof we need some preparations. First, we specify the
algebra $A^*$.
Denote by $\La^*(n)$ the subalgebra in $\C[x_1,\dots,x_n]$ formed by
the polynomials which are symmetric in `shifted' variables
$x'_j=x_j-j$, $j=1,\dots,n$. Define the projection map
$\Lt(n)\to\Lt(n-1)$ as the specialization $x_n=0$ and note that
this projection preserves the filtration defined by ordinary degree of
polynomials. Now we take the projective limit of $\La^*(n)$'s
in the category of filtered algebras as $n\to\infty$. The result is a
filtered algebra which is called the {\it algebra of shifted
symmetric functions\/} and denoted by $\La^*$.
The algebra $\La^*$ will be taken as the algebra $A^*$. As the
elements $\P^*_\mu$ we shall take the {\it shifted Schur functions\/}
$s^*_\mu$ as defined in \cite{OO}.
By the definition of $\La^*$, each element $f\in\La^*$ can be
evaluated at any sequence $x=(x_1,x_2, \dots)$ with finitely many
nonzero terms. In particular, we can evaluate shifted symmetric
functions at any $\la=(\la_1,\la_2,\dots)\in\Y$, which allows one to
interpret $\La^*$ as a certain algebra of functions on the Young
diagrams. This point of
view was developed in \cite{KO}. The shifted Schur
functions $s^*_\mu$ possess the Interpolation Property of \S1, see \cite{Ok1}, \cite{OO}.
For the one--row shifted Schur functions there is a special notation:
$h^*_m=s^*_{(m)}$. A useful tool is the following generating series for
the $h^*$ functions:
$$
H^*(u)=1+\sum_{m=1}^\infty \frac{h^*_m}{u(u-1)\dots(u-m+1)}\,.
\tag 2.5
$$
Here $u$ is a formal indeterminate and the series is viewed as an
element of $\La^*[[\frac1u]]$. Since the elements $h^*_m$ are
algebraically independent generators of $\La^*$, a multiplicative
functional $\pi:\La^*\to\C$ can be uniquely defined by assigning to
$H^*(u)$ an arbitrary formal power series in $\frac1u$ with constant
term 1. We shall use this fact below.
Note a useful formula
$$
H^*(u)(x_1,x_2,\dots)=\prod_{i=1}^\infty
\frac{u+i}{u+i-x_i}\,,
\tag 2.6
$$
see \cite{OO, Theorem 12.1}. Here, by definition,
$$
H^*(u)(x_1,x_2,\dots)=
1+\sum_{m=1}^\infty
\frac{h^*_m(x_1,x_2,\dots)}{u(u-1)\dots(u-m+1)}\,.
\tag 2.7
$$
The equality \tht{2.6} can be understood as
follows. We assume that only finitely many of $x_i$'s are distinct
from zero. Then the left--hand side, which is the series \tht{2.7},
converges in a left half--plane $\Re u<\operatorname{const}\ll0$ and
equals the right--hand side of \tht{2.6}.
For an element $f$ of $\La$ or $\La^*$, we shall
abbreviate
$$
f(x_1,\dots,x_k)=f(x_1,\dots,x_k,0,0,\dots).
$$
Recall the
combinatorial formula for the Schur functions:
$$
s_\mu(x_1,\dots,x_k)=\sum_T \prod_{b\in\mu}x_{T(b)}\,,
\tag 2.8
$$
where $T$ ranges over the set of Young tableaux of shape $\mu$
with entries in $\{1,\dots,k\}$, see \cite{M, I.5}. It
will be convenient for us to employ here the reverse tableaux (i.e.,
the entries $T(b)$ decrease from left to right along the rows and
down the columns). Since $s_{\mu}$ is symmetric, (2.8) also holds if
the sum in the right--hand side is taken over all reverse tableaux of
shape $\mu$ with entries in $\{1,\dots,k\}$.
We shall need a similar formula for the shifted
Schur functions:
$$
s^*_\mu(x_1,\dots,x_k)=\sum_T \prod_{b\in\mu}(x_{T(b)}-c(b))\,,
\tag 2.9
$$
where $T$ ranges over reverse tableaux of shape $\mu$ with entries in
$\{1,\dots,k\}$, see \cite{OO, Theorem 11.1}.
\proclaim{Proposition 2.2} Let $k=1,2,\dots$ and $z'\in\C$. The
following specialization formula holds
$$
s^*_\mu(\underbrace{-z',\dots,-z'}_k)=
(-1)^n\prod_{b\in\mu}
\frac{(k+c(b))(z'+c(b))}{a(b)+l(b)+1}\,,
\quad n=|\mu|.
\tag 2.10
$$
\endproclaim
\demo{Proof} Compare the combinatorial formulas \tht{2.8} and \tht{2.9}.
If $x_1=\dots=x_k=-z'$ then the product in \tht{2.9} does not
depend on $T$ and is equal to
$$
\prod_{b\in\mu}(-z'-c(b))
=(-1)^n\prod_{b\in\mu}(z'+c(b)).
$$
It follows that
$$
s^*_\mu(\underbrace{-z',\dots,-z'}_k)
=(-1)^n\prod_{b\in\mu}(z'+c(b))\cdot
s_\mu(\underbrace{1,\dots,1}_k).
$$
Now we apply the well--known specialization formula
$$
s_\mu(\underbrace{1,\dots,1}_k)
=\prod_{b\in\mu}\frac{k+c(b)}{a(b)+l(b)+1}\,,
$$
see \cite{M, I.3, Ex.4}, which implies \tht{2.10}. \qed
\enddemo
The argument used in the proof is borrowed from Okounkov's paper
\cite{Ok4}, the derivation of formula (1.9); see also Proposition 3.2 below.
\proclaim{Corollary 2.3} For any $z,z'\in\C$, the linear functional
$\pi_{zz'}:\La^*\to\C$ given by
$$
\pi_{zz'}(s^*_\mu)=
(-1)^n\prod_{b\in\mu}
\frac{(z+c(b))(z'+c(b))}{a(b)+l(b)+1},
\quad n=|\mu|,
\tag 2.11
$$
is multiplicative.
\endproclaim
\demo{Proof} Indeed, this expression depends polynomially on $z$. So,
it suffices to prove the multiplicativity of $\pi_{zz'}$ in
the case $z=k$, where $k=1,2,\dots$ . By Proposition 2.2, in this
case our functional is the evaluation at the point
$$
x=(\underbrace{-z',\dots,-z'}_k,0,0,\dots).
$$
Consequently, the functional is multiplicative. \qed
\enddemo
\demo{Proof of Theorem 2.1} We apply Proposition 1.3 by taking
$A^*=\La^*$ and $\P^*_\mu=s^*_\mu$. The Pieri--type formula for
$s^*$--functions ([OO, Theorem 9.1]) shows that the relation
\tht{1.8}
holds with the sequence $a_n=n$. We take as $\pi$ the multiplicative
functional $\pi_{zz'}$ afforded by Corollary 2.3. It follows from
\tht{2.11} that
$$
-\pi_{zz'}(s^*_{(1)})=zz'=t,
$$
so that we may substitute $t$ into \tht{1.11}. Finally, the condition
\tht{1.10} is just the assumption on $t$ given in Theorem 2.1. Thus, the
expression \tht{2.4} is a special case of \tht{1.11}, which concludes the
proof. \qed
\enddemo
\example{Remark 2.4} In terms of the generating series \tht{2.5} for the
$h^*$ functions, the multiplicative functional $\pi_{zz'}$ can be
described as follows:
$$
\gathered
\pi_{zz'}(H^*(u))=1+\sum_{m=1}^\infty\frac{(z)_m(z')_m}{(-u)_m\,m!}\\
={}_2F_1(z,z';-u;1)
=\frac{\Ga(-u)\Ga(-u-z-z')}{\Ga(-u-z)\Ga(-u-z')}\,,
\endgathered
\tag 2.12
$$
where we assume $\Re u\ll0$; the last equality follows from Gauss'
summation formula.
\endexample
\proclaim{Proposition 2.5} The function $\varphi_{zz'}$ afforded by
Theorem 2.1 is a nondegenerate function from $\Harm^+_1(\Y)$ if and
only if the parameters satisfy one of the following two conditions:
$\bullet$ either $z'=\bar z$ where $z\in\C\setminus\Z$,
$\bullet$ or $z,z'$ are real and there exists $m\in\Z$ such that
$m0)=(1^{r_1(\mu)}2^{r_2(\mu)}\dots)
$$
denote an arbitrary partition also viewed as a Young diagram.
In this section we are dealing with the monomial symmetric functions
$m_\mu$ \cite{M, I.2}. They form a basis of the algebra
$\La$ and obey the relation:
$$
m_\mu m_{(1)}=\sum_{\la:\, \la\searrow\mu} \ka_0(\mu,\la)m_\la\,,
\tag 4.1
$$
where the positive integers $\ka_0(\mu,\la)$ are defined as follows:
if $k$ stands for the length of the
row in $\la$ containing the box $\la\setminus\mu$ then
$\ka_0(\mu,\la)=r_k(\la)$.
The {\it Kingman graph\/} $\K$ is the multiplicative graph associated
with the algebra $\La$ and its basis $\{m_\mu\}$ \cite{Ke1}. I.e., this is the
Young graph with the formal edge multiplicities $\ka_0(\mu,\la)$. Since
the numbers $\ka_0(\mu,\la)$ are integers, one can regard $\K$ as a
graph with multiple edges.
Next, introduce the {\it factorial monomial symmetric functions\/}
$m^*_\mu$, which are also elements of $\La$. By definition
\cite{Ke1}, $m^*_\mu$ is the sum of all
{\it distinct\/} expressions obtained from
$$
\prod_{i=1}^l x_i(x_i-1)\dots(x_i-\mu_i+1)
$$
by permutations of the variables $x_1,x_2,\dots$ . Thus, the definition of $m^*_\mu$ is similar to that of $m_\mu$, the only difference
is that the ordinary powers $x^m$ are replaced by the
falling factorial powers $x(x-1)\dots(x-m+1)$.
The function $m^*_\mu$ can be characterized as the only symmetric function
with the highest term $m_\mu$ and such that
$m^*_\mu(\la_1,\la_2,\dots)=0$ for any diagram $\la\ne\mu$,
$|\la|\le|\mu|$. Thus, $m^*_\mu$ possesses the Interpolation Property
of \S1.
One can directly verify that
$$
m^*_\mu m^*_{(1)}=n m^*_\mu+\sum_{\la:\, \la\searrow\mu}
\ka_0(\mu,\la)m^*_\la, \qquad n=|\mu|
\tag 4.2
$$
(this also follows from the Interpolation Property).
\proclaim{Theorem 4.1} Let $t,\al$ be complex parameters,
$t\ne-1,-2,\dots$\,.
Then the function
$$
\varphi_{t,\al}(\mu)=
\frac{(\mu_1-1)!\dots(\mu_l-1)!}{r_1(\mu)!r_2(\mu)!\dots}\cdot
\frac{t(t+\al)\dots(t+(l-1)\al)}{(t)_n}\cdot
\prod\Sb b=(i,j)\in\mu\\j\ge2\endSb\left(1-\frac{\al}{j-1}\right)\,, \tag 4.3
$$
where $n=|\mu|$, is harmonic on the graph $\K$. Here $l$ is the length (number of nonzero parts) of $\mu$.
The functions \tht{4.3} fit into the general scheme of Proposition 1.3 with
$A^*=\La$ and $\P^*_\mu=m^*_\mu$.
\endproclaim
As is explained below, the first claim is equivalent to a result of
Pitman \cite{Pi}.
\demo{Proof} According to Proposition 1.3,
it suffices to check that
there exists a multiplicative functional $\pi_{t,\al}: \La\to\C$ such
that
$$
\gathered
\pi_{t,\al}(m^*_\mu)=(-1)^n\,
\frac{(\mu_1-1)!\dots(\mu_l-1)!}{r_1(\mu)!r_2(\mu)!\dots}\\ \cdot\,
t(t+\al)\dots(t+(l-1)\al)\cdot
\prod\Sb b=(i,j)\in\mu\\j\ge2\endSb\left(1-\frac{\al}{j-1}\right)\,. \endaligned
\tag 4.4
$$
As the functions $m^*_\mu$ form a basis in $\La$, we can define a
linear functional $\pi_{t,\al}:\La\to\C$ by the formula \tht{4.4}. We
claim that it is multiplicative if $t=-k\al$, where $k=1,2,\dots$\,.
To see this we shall prove that $\pi_{-k\al,\al}$ coincides with the
specialization at the point $(\underbrace{\al, \dots,\al}_k)$.
Indeed, from the definition of $m^*_\mu$ it follows that
$$
m^*_\mu(\underbrace{\al,\dots,\al}_k)
=\frac{k(k-1)\dots(k-l+1)}{r_1(\mu)!r_2(\mu)!\dots}\,
\prod_{i=1}^l \al(\al-1)\dots(\al-\mu_i+1),
\tag 4.5
$$
and a direct verification shows that this expression coincides with
$\pi_{-k\al,\al}(m^*_\mu)$.
Finally, as the right--hand side of \tht{4.4} depends on the
parameters $t,\al$ polynomially, $\pi_{t,\al}$ is multiplicative for
all values of the parameters. \qed
\enddemo
\proclaim{Proposition 4.2} The function $\varphi_{t,\al}$ afforded by
Theorem 4.1 is a nondegenerate function from $\Harm^+_1(\K)$ if and
only if the parameters $t,\al$ are real and satisfy the inequalities
$0\le\al<1$, $t>-\al$.
\endproclaim
The proof is straightforward.
There is a bijective correspondence between the functions
$\varphi\in\Harm^+_1(\K)$ and the {\it partition structures\/} in the
sense of Kingman \cite{Ki1}, \cite{Ki2}. According to Kingman, a partition
structure is a sequence $M=(M_n)$ of probability distributions on
the partitions of $n$, $n=1,2,\dots$, such that for each $n$, $M_n$
and $M_{n+1}$ are connected by a certain consistency relation. These
sequences are nothing else than the sequences $(M_n)$ as defined in
\S1, see (1.4). Thus, the passage from a harmonic function
$\varphi\in\Harm^+_1(\K)$ to the corresponding partition structure
is given by the formula
$$
M_n(\mu)=\varphi(\mu)\cdot\dim_\K(\mu), \qquad |\mu|=n,
$$
where
$$
\dim_\K(\mu)=\frac{n!}{\mu_1!\dots\mu_l!}
\tag 4.6
$$
(the latter formula readily follows from the general relation
\tht{1.12} if we substitute $\lambda=\mu$ and $\Cal
P^*_\mu=m^*_\mu$).
Under this correspondence, the functions $\varphi_{t,0}$ with $t>0$ turn
into Ewens' partition structures \cite{Ew}. More general
functions $\varphi_{t,\al}$ with the restrictions $t>-\al$, $0\le\al<1$,
of Proposition 4.2 correspond to Pitman's two--parameter
generalization of Ewens' partition structures
\cite{Pi}, \cite{Ke4}. Note that the harmonic functions $\varphi_{-k\al,\al}$ with
$k=1,2,\dots$, which appear in the proof of Theorem 4.1, are nonnegative and
{\it degenerate\/} provided that $\al<0$; the significance of the
corresponding partition structures is explained in the introduction
to \cite{Pi}.
Here is yet another interpretation of the harmonic functions $\varphi_{t,\al}$.
For $n\ge2$ there exists a unique map $S(n)\to S(n-1)$
which commutes with the two--sided action of the smaller group
$S(n-1)$. This map, called the {\it canonical projection,\/} can be
defined as follows: if $s,s_1,s_2\in S(n-1)$ and $(n-1,n)$ stands for
the elementary transposition of ``$n-1$'' and ``$n$'', then
$s\mapsto s$ and $s_1\cdot(n-1,n)\cdot s_2\mapsto s_1s_2$. In other
words, the canonical projection is defined by removing ``$n$'' from
the cycle $\dots i\to n\to j\dots$ containing it, see \cite{KOV}.
The projective limit $\X=\varprojlim S(n)$ taken with respect to the
canonical projections is a compact topological space. Its elements are
called {\it virtual permutations.\/} There is a natural
embedding $S(\infty)\to\X$ whose image is dense, so that $X$ is a
certain compactification of the discrete set $S(\infty)$. The
two--sided action of $S(\infty)$ on itself can be extended to $\X$,
which makes $\X$ a $S(\infty)\times S(\infty)$-space. This
construction and its meaning for the representation theory of the
group $S(\infty)$ is discussed in \cite{KOV}.
A probability measure $\M$ on $\X$ is called {\it central\/} if it is
invariant under the action of the diagonal subgroup in
$S(\infty)\times S(\infty)$ (that action extends the action of
$S(\infty)$ on itself by conjugations).
There is a natural bijective correspondence
$\varphi\leftrightarrow \M$ between the elements
$\varphi\in\Harm^+_1(\K)$ and the central measures $\M$ on $\X$. It is
specified as follows: for each $n=1,2,\dots$, the image of $\M$ under
the composite projection $\X\to S(n)\to\K_n$ coincides with $M_n$; here
the second arrow $S(n)\to\K_n$ assigns to a permutation its cycle
structure which is identified with a partition, and $(M_n)$ is the
partition structure corresponding to $\varphi$.
Let us denote by $\M_{t,\al}$ the measures corresponding to the
harmonic functions $\varphi_{t,\al}$, where the parameters satisfy the
conditions of Proposition 4.2. The measure $\M_{1,0}$ is invariant with
respect to $S(\infty)\times S(\infty)$ and it is the only probability
measure with this property. The measures $\M_{t,0}$ with $t>0$ were
employed in \cite{KOV} for a geometric construction of generalized regular representations of the group $S(\infty)\times S(\infty)$ which are closely related to the harmonic functions $\varphi_{zz'}$ defined in \S2. Note that all the measures $\M_{t,\al}$ are quasi--invariant under the action of this group, see \cite{KOV}, \cite{Ke4}.
Extreme points of the convex set $\Harm^+_1(\K)$ can be parametrized
by nonincreasing sequences of nonnegative numbers with sum less
or equal to 1 (cf. \tht{2.15}):
$$
\Om(\K)=\{\al_1\ge\al_2\ge\ldots\,|\, \sum_{i}\al_i\le 1\},
\tag 4.7
$$
and the Poisson kernel $K(\mu,\omega)$ is given in this case
by {\it extended monomial symmetric functions}:
$$
K(\mu,\al)=K(1^{r_1}2^{r_2}\dots,\al)=
\sum_{k=0}^{r_1}\frac{(1-\sum_i\al_i)^k}{k!}\,
m_{(1^{r_1-k}2^{r_2}\dots)}(\al_1,\al_2,\dots),
\tag 4.8
$$
see \cite{Ki2}, \cite{Ke4}.
Let us also note that the Kingman graph
may be viewed as the degeneration of the Jack graph $\J$ as
$\th\to0$. Indeed, according to the definition of the Jack functions,
their expansion in the monomial functions has the form
$$
P_\mu=m_\mu\,+\,\text{lower terms}
$$
relative to the dominance order on partitions \cite{M, VI, (10.13)}. It is well
known that in the limit $\th\to 0$ the coefficients of all the lower
terms vanish. In this sense, the Jack functions $P_\mu$
degenerate to the monomial functions $m_\mu$ as $\th\to0$. This
implies, in particular, the limit relation
$$
\ka_0(\mu,\la)=\lim_{\th\to0}\ka_\th(\mu,\la),
$$
which can also be checked directly from \tht{3.2}.
\head \S5. The Schur graph \endhead
Recall that a partition is said to be {\it strict\/} if its nonzero
parts are pairwise distinct. In this section, the symbols $\mu$
and $\la$ always mean strict partitions. Using the standard
correspondence between partitions and Young diagrams we introduce the
relation $\mu\nearrow\la$ as before, see \S2. Then the {\it Schur graph\/}
$\SS$ is defined as follows: the vertices of the
$n$th floor $\SS_n$ are the strict partitions of $n$, and the
edges are the couples $\mu\nearrow\la$. By definition, the empty
partition $\varnothing$ is included into the set of the strict
partitions. All the edge multiplicities are equal to 1. Thus, the
Schur graph is a subgraph of the Young graph.
Let $\Ga$ denote the subalgebra in $\La$ generated by the odd power
sums $p_1=\sum_i x_i$, $p_3=\sum_i x_i^3,\dots$\,. Equivalently, $\Ga$
consists of those symmetric functions $f(x_1,x_2,\dots)$ which
satisfy the following cancellation condition: for any $i\ne j$, the
result of specializing $x_i=y$, $x_j=-y$ in $f$ does not depend on
$y$ (see \cite{Pr}).
In this section, the symbol $P_\mu$ stands for the Schur $P$ function
indexed by a strict partition $\mu$. The Schur $P$ functions form a
homogeneous basis of $\Ga$, $\deg P_\mu=|\mu|$. They obey the
following Pieri--type rule:
$$
P_\mu P_{(1)}=\sum_{\la:\,\la\searrow\mu} P_\la\,,
\tag 5.1
$$
see \cite{M, III.8}. Thus, $\SS$ is a multiplicative graph with $A=\Ga$ and
$\{\P_\mu\}=\{P_\mu\}$.
Note that
$$
\dim_\SS\mu=\frac{n!}{\prod_{i=1}^{\ell(\mu)}\mu_i!}
\prod_{1\le i0$. The
following expression is a positive harmonic function on the Schur graph: $$
\varphi_t(\mu)=\frac1{(t)_n}\,
\frac{\prod_{(i,j)\in\mu} (2t+(j-1)j)}
{2^{\ell(\mu)}\,\prod_{i=1}^{\ell(\mu)}\mu_i!} \,
\prod_{1\le iq\ge1$ to a larger set of indices by adopting the convention
$P^*_{(p,q)}=-P^*_{(q,p)}$. Thus, $P^*_{(p,q)}$ makes sense for any $p,q=1,2,\dots$ (we assume that $P^*_{(p,q)}=0$ for $p=q$). Then we
have two families of relations:
$$
\gather
P^*_{(p,1)}=P^*_{(p)}P^*_{(1)}-pP^*_{(p)}\,,
\tag 5.15
\\
P^*_{(p+1,q)}+P^*_{(p,q+1)}+(p+q)P^*_{(p,q)}=
P^*_{(p)}P^*_{(q+1)}-P^*_{(p+1)}P^*_{(q)}-(p-q)P^*_{(p)}P^*_{(q)}\,.
\tag 5.16
\endgather
$$
Here $p,q$ range over $\{1,2,\dots\}$. These relations were proved in
\cite{I2}. Note that \tht{5.15} can be formally obtained from
\tht{5.16} by substituting $q=0$.
Using double
induction on $p+q$ and $q$ we see that these relations indeed
allow to express the two--row functions through the one--row
functions.
A direct computation shows that the relations \tht{5.15}, \tht{5.16}
remain valid if we apply $\pi_t$ to each $P^*$ function involved. This means that \tht{5.12} holds on the two--row functions.
Finally, to handle arbitrary $P^*$ functions we employ the following
relation proved in \cite{I2}:
$$
P^*_\mu
=\Pf\left[P^*_{(\mu_i,\mu_j)}\right]_{1\le i,j\le\ell(\mu)+\varepsilon}
\,.
\tag 5.17
$$
Here the symbol $\Pf$ means Pfaffian and $\varepsilon$ equals 1 for
odd $\ell(\mu)$ and 0 for even $\ell(\mu)$, so that the order of the matrix
is always even. Note that this relation has exactly the same form as
in the case of the classical Schur $P$ functions, see \cite{M, III.8}.
To conclude that \tht{5.12} holds on any $P^*_\mu$ we must verify that
\tht{5.17} remains valid if we apply $\pi_t$ to each $P^*$ function. This
readily follows from the well--known relation
$$
\prod_{1\le i0$.
\endproclaim
The proof is straightforward.
It should be noted that the values $t=k(1-k)/2$ used in the proof of
Theorem 5.1 lie outside the region $t>0$.
Quite similarly to the case of the Young graph, extreme points of
$\Harm_1^+(\SS)$ correspond to {\it projective characters} or {\it
projective finite factor representations} of the group $S(\infty)$.
They can be parametrized by the points of the infinite--dimensional
simplex $\Om(\SS)=\Om(\K)$ described by (4.7), see \cite{N}, \cite{I1}.
The Poisson kernel $K(\mu,\omega)$ for the Schur graph is given by
the image of the Schur $P$ function $P_\mu$ under the specialization
of the algebra $\Gamma$ which sends odd powers sums to the following
expressions (cf. (2.16), (3.9)) $$
(x_1^k+x_2^k+\ldots)\mapsto\cases 1,& k=1,\\
\sum_{i=1}^\infty\alpha_i^k,& k=2m+1\ge 3.
\endcases
\tag 5.19
$$
\head \S6. Finite--dimensional specializations
\endhead
In the previous four sections we described four different examples of
graphs which fit into the general scheme introduced in \S1. For each
of these graphs we produced a nontrivial family of
specializations of the corresponding algebras $A^*$ which defined,
according to Proposition 1.3, a certain family of (nonnegative)
harmonic functions on the graph.
Every such family, in its turn, gives rise to a family of probability
measures on the space $\Omega(\G)$ (Theorem 1.1), and this space in
all our examples is infinite--dimensional, see (2.15), (4.7).
For Young and Kingman graphs such measures have been thoroughly
studied, see \cite{P.I--P.V}, \cite{BO1}, \cite{BO2}, \cite{Ki3}, \cite{Pi}, \cite{Ke4}. They lead to certain stochastic
processes on the real line, for the Young graph the processes are
closely related to those arising in Random Matrix Theory, while for
the Kingman graph the theory is connected with Poisson processes.
Our goal in this section is to construct `simpler' families of
harmonic functions for Young, Kingman, and Schur graphs. The word
`simpler' means that the corresponding measures on $\Omega(\G)$ will
be supported by finite--dimensional subspaces. These measures will be
explicitly computed.
The corresponding Poisson integrals (which arise due to Theorem 1.1)
will give (possibly new) integral formulas involving products of Schur $S$ and $P$ functions, see
\tht{6.10}, \tht{6.15}, \tht{6.23}, \tht{6.26} below.
\subhead 6.1. Truncated Young branching \endsubhead Recall that for
the Young graph $\Y$ the algebra $A^*$ is the algebra $\La^*$ of
functions in infinitely many variables $x_1,x_2,\ldots$ symmetric in
`shifted' variables $x_j'=x_j-j$.
We shall consider the most natural specializations of this algebra
obtained by fixing finitely many variables $x_1,\ldots,x_l$ and
sending remaining variables $x_{l+1},x_{l+2},\ldots$ to zero.
In any such specialization all functions $s_\mu^*$ with the length (number of nonzero parts) of $\mu$ greater than $l$ vanish, see \cite{OO}. This implies that the harmonic function on $\Y$ afforded by Proposition 1.3 vanishes on all Young diagrams with more than $l$ rows. Thus, one can consider such a function as a harmonic function on the subgraph $\Y(l)$ of $\Y$ consisting of all Young diagrams with length $\le l$. This graph fits into the general formalism of $\S1$: the algebra $A$ is the algebra of symmetric polynomials in $l$ variables, and the algebra $A^*$ is the algebra of shifted symmetric polynomials in $l$ variables. The elements $\P_\mu$ and $\P_\mu^*$ are conventional and shifted Schur polynomials in $l$ variables, respectively. The graph $\Y(l)$ is called the {\it truncated Young graph}. Harmonic functions on such graphs were considered by Kerov, see \cite{Ke3}; he used them to derive certain Selberg--type integrals. Our arguments below are similar to those of Kerov's work.
Let us fix a Young diagram $\lambda$. Denote by $l$ the length of $\lambda$. We shall assume that $l\ge 2$. Denote by $\pi_\lambda$ the algebra homomorphism
$\pi_\lambda:\Lambda^*\to\R$ defined by
$$
\cases
x_i\mapsto-\la_i-2(l-i)-1, &1\le i\le l,\\
x_i \mapsto 0 &i>l.
\endcases
\tag 6.1
$$
Convenience of such choice of notation will be clear in a while.
According to the general scheme of \S1 (Proposition 1.3), the
corresponding harmonic function on $\Y$ has the form
$$
\varphi_{\lambda}(\mu)=
\frac{(-1)^{|\mu|}\pi_\la(s^*_\mu)}{(-\pi_\la(s^*_{(1)}))_{|\mu|}}=
\frac{(-1)^{|\mu|}s^*_\mu(-\lambda-2\delta-1)}{(|\lambda|+l^2)_{|\mu|}},
\tag 6.2
$$
where
$s^*_\mu$ is the shifted Schur function and
$$
\cases
\delta_i=l-i, &1\le i\le l,\\
\delta_i=0, &i>l.
\endcases
$$
\proclaim{Proposition 6.1} The function $\varphi_\lambda$ defined by
\tht{6.2} is nonnegative.
\endproclaim
\demo{Proof} Let the symbol $(a\dhr k)$ denote the {\it $k$th falling factorial
power of $a$}:
$$
(a\dhr k)=\cases a(a-1)\cdots (a-k+1),&k=1,2,\dots\,,\\ 1,&k=0.
\endcases
$$
Recall the definition of the shifted Schur polynomials in finitely many
variables \cite{OO}
$$
s_\mu^*(x_1,\ldots,x_l)=
\frac{\det[(x_i+\delta_i\dhr \mu_j+\delta_j)]_{i,j=1}^l}
{\det[(x_i+\delta_i\dhr \delta_j)]_{i,j=1}^l}\,,
\tag 6.3
$$
where $\delta_j$ are as above.
Let us plug in our $x_i$ from \tht{6.1} to
\tht{6.3}. We get
$$
\gathered
(-1)^{|\mu|}\pi_{\lambda}(s^*_\mu)=
(-1)^{|\mu|}\,
\frac{\det[(-\lambda_i-\delta_i-1\dhr \mu_j+\delta_j)]_{i,j=1}^l}
{\det[(-\lambda_i-\delta_i-1\dhr \delta_j)]_{i,j=1}^l}\\=
\frac{\det[\Gamma(\lambda_i+\delta_i+1+\mu_j+\delta_j)]_{i,j=1}^l}
{\det[\Gamma(\lambda_i+\delta_i+1+\delta_j)]_{i,j=1}^l}.
\endgathered
$$
As the
first $l$ members of sequences
$\lambda+\delta+1$, $\mu+\delta$, and $\delta$ decrease,
the claim follows from the inequality
$$
\det[\Gamma(x_i+y_j)]_{i,j=1}^l>0, \qquad x_1>\dots>x_l> 0,\quad y_1>\dots>y_l> 0,
$$
which is a special case of Problem VII.66 in \cite{PS}. \qed
\enddemo
Next, we form the probability distributions $M_n=M_n^\la$ on $\Y_n$ for each $n=0,1,2,\dots$ according to (1.4):
$$
M_n^\la(\nu)=\dim_\Y(\nu)\,\varphi_\la(\nu),\qquad |\nu|=n.
\tag 6.4
$$
Let us embed $\Y_n \hookrightarrow \Om(\Y)$ via
$$
\nu\mapsto \left(\frac {\nu_1}{n},\frac{\nu_2}{n},\dots;\,0,0,\dots\right)
\in \Om(\Y).
\tag 6.5
$$
As was mentioned in \S1, the probability measure $P$ involved in the Poisson integral (1.2) is the weak limit of the images of the measures $M_n$ under appropriate embeddings $\Y_n \hookrightarrow \Om(\Y)$; as such embeddings one can take (6.5).
Let us denote by $V(a)$ the Vandermonde determinant
$$
V(a_1,\dots,a_s)=\prod_{1\le i0$ is arbitrary, then, as $n\to\infty$,
$$
M_n^{\lambda}(\nu)=
n^{1-l}\,\frac{\Gamma(|\lambda|+l^2)\,
s_\lambda(\al_1,\dots,\al_l)V^2(\al_1,\ldots,\al_l)}
{\det[\Gamma(\lambda_i+\delta_i+\delta_j+1)]_{i,j=1}^l}\,(1+o(1))
\tag 6.7
$$
with
$$
\alpha_1=\frac{\nu_1}n,\dots, \alpha_l=
\frac{\nu_l}n,
$$
and the estimate \tht{6.7} is uniform in $\nu$.
\endproclaim
Let us postpone the proof of this statement and proceed with the
proof of Proposition 6.2 taking Lemma 6.3 for granted.
The Thoma simplex $\Om(\Y)$ defined in (2.15) is a compact topological space. Let $C(\Om(\Y))$ denote the algebra of continuous functions on $\Om(\Y)$. Take any $f\in C(\Om(\Y))$. Note that $f$ is bounded. The value on $f$ of the image of $M_n^{\lambda}$ under the $n$th embedding (6.5) have the form
$$
\sum_{\nu\in \Y_n}M_n^{\lambda}(\nu)\cdot
f\left(\frac{\nu_1}n, \frac{\nu_2}n,\ldots;\,
0,0,\ldots\right).
\tag 6.8
$$
Recall that $M_n^{\lambda}$ is supported by the Young diagrams $\nu$ with the length $\le l$. Let us first consider the part of
the sum (6.8) involving diagrams $\nu$ with $\nu_l\ge \varepsilon n$, $\nu_{l+1}=0$. Using Lemma 6.3 and the boundedness of $f$, we get
$$
\gathered
\sum_{\Sb \nu\in \Y_n\\
\nu_l\ge\varepsilon n,\,\nu_{l+1}=0\endSb} M_n^{\lambda}(\nu)\cdot f\left(\frac{\nu_1}n, \frac{\nu_2}n,\ldots;\,
0,0,\ldots\right)
= \frac{{\Gamma(|\lambda|+l^2)}\,(1+o(1))}
{\det[\Gamma(\lambda_i+\delta_i+\delta_j+1)]_{i,j=1}^l}
\\ \times\sum_{\Sb \nu\in \Y_n\\
\nu_l\ge \varepsilon n,\, \nu_{l+1}=0\endSb}f(\al_1,\dots,\al_l,0,\dots;0,0,\dots)\cdot s_\lambda(\al)V^2(\al)\cdot n^{1-l}\,,
\endgathered
\tag 6.9
$$
where $\al=(\al_1,\dots,\al_l)$ is as in Lemma 6.3.
The sum in the right--hand side of (6.9) is a Riemannian sum for the
integral
$$
\int f(\al_1,\dots,\al_l,0,\dots;0,0,\dots)\cdot s_\lambda(\alpha)V^2(\alpha)\,d\alpha
$$
over the part of $\Delta_l$ specified by the condition $\al_l\ge \varepsilon$.
Thus, it remains to prove that the part of the sum (6.8) involving
diagrams $\nu$ with $\nu_l<\varepsilon n$ is $\varepsilon$--negligible.
Since $M_n^\la(\,\cdot\,)=O(n^{1-l})$, this follows from the fact that the number of Young diagrams $\nu=(\nu_1,\dots,\nu_l,0,\dots)$ with $\nu_1+\dots+\nu_l=n$ and such that $\nu_l<\varepsilon n$ is bounded by $const\cdot \varepsilon\, n^{l-1}$. This completes the proof of Proposition 6.2 modulo Lemma 6.3.
\enddemo
\demo{Proof of Lemma 6.3} We shall employ formulas (6.2), (6.3),
(6.4).
Denote by $m$ the length of $\nu$.
We apply a well--known dimension formula
$$
\dim_{\Y}\nu=\frac{n!}{\prod_{i=1}^{m}(\nu_i+m-i)!}
\prod_{1\le im$.
Next, we have the following asymptotic relations as $n\to\infty$:
$$
\prod_{1\le i0)$
by any positive real numbers satisfying the same system of
inequalities. Then the Schur function $s_\la(\al)$ should be understood
just as the ratio
${\det [\al_i^{\la_j+l-j}]}/{\det[\al_i^{l-j}]}.$ We restricted ourselves to integral $\la_i$'s in order to emphasize the symmetry $\la\leftrightarrow\mu$ in the integral (6.10).
By analytic continuation, the formula (6.10) can be extended to arbitrary complex $\la_i$'s. When $\la$ has the form $((l-1)\theta+a,(l-2)\theta+a,\dots,a)$, where $\theta>0$ and $a>-1$, the measure (6.6) is related to the so--called Laguerre biorthogonal ensemble, see \cite{B2}.
\endexample
\subhead 6.2. $\Gamma$--shaped Young branching \endsubhead
As in the previous subsection, we work with the Young graph $\Y$.
This time we shall use another, so--called super realization of the algebra
$A^*=\Lambda^*$ of shifted symmetric functions. Since a detailed exposition of the material of this subsection would be rather tedious, we shall only state the results and outline the ideas used in the proofs.
Let $\wt \La$ be the algebra of supersymmetric functions in
$x=(x_1,x_2,\dots)$ and $y=(y_1,y_2,\dots)$, see \cite{BR}, \cite{M,
Ex.I.3.23-24, Ex.I.5.23}. It can be identified with the algebra $\La$
of symmetric functions; under this identification power sums
$p_m\in\Lambda$ correspond to their super analogs
$$
p_m(x;y)=\sum_ix_i^m+(-1)^{m-1}\sum_iy_i^m.
$$
Note that our notation slightly differs from that of Macdonald's
book: his supersymmetric functions in $x$ and $y$ coincide with ours in $x$ and $-y$.
Below we shall use Frobenius notation for Young diagrams, its description can be found in \cite{M, I.1}.
\proclaim{Theorem 6.6 \cite{KO}} There exists an algebra isomorphism
$\rho:\La^*\to \wt \La$ such that for any $f\in\La^*$ and any Young
diagram $\la=(\la_1,\la_2,\dots)$ with Frobenius coordinates
$(p_1,\ldots p_d|\, q_1,\ldots,q_d)$ the following equality holds:
$$
f(\la_1,\la_2,\dots)=\rho(f)\left({p_1+\frac
12},\ldots,{p_d+\frac12};\, {q_1+\frac12},\ldots,{q_d+\frac12}\right).
$$
\endproclaim
Now we shall identify $\wt\La$ and $\La^*$ using the isomorphism $\rho$. We shall denote the elements $\rho(s^*_\mu)\in\wt\Lambda$ as $FS_\mu$ and call them {\it Frobenius--Schur functions}, see \cite{ORV}.
Let us consider
specializations of the algebra $\La^*\simeq\La$ obtained by fixing finitely many variables $x_1,\dots,x_d;\, y_1,\dots,y_d$
and sending remaining variables $x_i,y_i,\ i=d,d+1,\dots$, to zero. In any such specialization all functions $s^*_\mu$ with the depth (number of diagonal boxes) greater than $d$ vanish, this follows from results of \cite{ORV}. Hence, the corresponding harmonic functions are concentrated on a subgraph of the Young graph $\Y$ consisting of diagrams with depth $\le d$. These are exactly the Young diagrams which fit into the $\Gamma$--shaped figure with $d$ rows and $d$ columns. We denote the subgraph of such Young diagrams by $\Y(d,d)$ and call it the {\it $\Gamma$--shaped Young graph}. Like the truncated Young graphs considered in 6.1, the $\Gamma$--shaped Young graphs also fit into the general formalism of \S1. The algebras $A$ and $A^*$ are both identified with the algebra of supersymmetric polynomials in $d+d$ variables, the elements $\P_\mu$ are supersymmetric Schur polynomials, and the elements
$\P_\mu^*$ are supersymmetric Frobenius--Schur polynomials.
Fix a Young diagram $\lambda$ with Frobenius coordinates
$(p_1,\ldots p_d\,|\, q_1,\ldots,q_d)$. We shall denote by $\pi_\la$
the algebra homomorphism $\pi_\la:\La^*\to\R$ defined by
$$
\cases
x_i\mapsto -p_i-\frac 12,&1\le i\le d,\\
y_i\mapsto -q_i-\frac 12,&1\le i\le d,\\
x_i,\,y_i\mapsto 0,&i>d.
\endcases
\tag 6.11
$$
According to \S1, the harmonic function on $\Y$ corresponding to
$\pi_\la$ has the form
$$
\varphi_\la(\mu)=\frac{(-1)^{|\mu|}\pi_\la(s^*_\mu)}{(-\pi_\la(s^*_{(1)}))_{|\mu|}}=\frac{(-1)^{|\mu|}FS_\mu(-p-\frac 12;-q-\frac
12)} {(|\la|)_{|\mu|}}.
\tag 6.12
$$
Unfortunately, it is not clear how to prove directly that $\varphi_\la$ is nonnegative. However, we can go around this obstacle.
Set
$$
M_n^\la(\nu)=\dim_\Y(\nu)\,\varphi_\la(\nu),\qquad |\nu|=n.
$$
Consider the embeddings $\Y_n \hookrightarrow \Om(\Y)$ defined as follows. For a Young diagram $\nu\in\Y_n$ with Frobenius coordinates $(P_1,\dots,P_D|\,Q_1,\dots, Q_D)$
$$
\nu \mapsto \left(\frac{P_1+1/2}n,\dots,\frac{P_D+1/2}n,0,\dots;\,\frac{Q_1+1/2}n,\dots,\frac{Q_D+1/2}n,0,\dots\right)\in\Om(\Y).
\tag 6.13
$$
\proclaim{Proposition 6.7} The images of (possibly signed) measures $M_n^\la$ under the embeddings \tht{6.13} weakly converge, as $n\to\infty$, to a (positive) probability measure $P$ on $\Om(\Y)$. This measure is supported by the finite--dimensional face
$$
\gather
\Delta_{d,d}=\{(\alpha,\beta)\in\Om(\Y)\,|\,\al_{d+1}=
\be_{d+1}=\al_{d+2}=\be_{d+2}=\dots=0,\
\sum_{i=1}^d(\al_i+\be_i)=1\}\\ \backsimeq
\{(\al_1,\ldots,\al_d;\be_1,\dots,\be_d)
\in \Bbb R^{2d}_+|\al_1\ge\dots\ge \al_d;\,
\be_1\ge\dots\ge \be_d,\ \sum_{i=1}^d(\al_i+\be_i)=1 \},
\endgather
$$
and its density with respect to the Lebesgue measure $d(\al;\be)$ on $\Delta_{d,d}$ equals
$$
\frac{\Gamma(|\lambda|)}
{\prod_{i=1}^d p_i!q_i!}\,
\left[{\det}\left(\frac 1{p_i+q_j+1}\right)\right]^{-1}\,
s_\lambda(\alpha;\beta)
\,{\det}^2\left(\frac 1{\alpha_i+\beta_j}\right),
\tag 6.14
$$
where $s_\la(\al;\be)$ is the supersymmetric Schur polynomial in $d+d$ variables.
\endproclaim
The proof of this proposition is quite similar to that of Proposition 6.2. An analog of Lemma 6.3 is proved using the Sergeev--Pragacz formula for $s_\la(\al;\be)$ (see \cite{PT}, \cite{M, I.3, Ex.24}) and its analog for the Frobenius--Schur polynomials (see \cite{ORV}).
It turns out that Proposition 6.7 implies the existence of the Poisson integral representation (1.2) for the harmonic function (6.12). The proof of this claim is quite similar to the proof of Theorem B in \cite{KOO}. Since the Poisson kernel is always nonnegative, the existence of the Poisson integral representation implies that our harmonic function is nonnegative.
Explicitly, the
Poisson integral representation has
the following form, cf. Proposition 6.4.
\proclaim{Proposition 6.8}
$$
\gathered
\frac{(-1)^{|\mu|}FS_\mu(-p-\frac 12;-q-\frac 12)}{(|\la|)_{|\mu|}}=
\frac{\Gamma(|\lambda|)}{\prod_{i=1}^d p_i!q_i!}\,
\left[{\det}\left(\frac 1{p_i+q_j+1}\right)\right]^{-1}\\
\times \int_{\Delta_{d,d}}s_{\mu}(\alpha;\be)\,
s_{\lambda}(\alpha;\be)\,{\det}^2\left(\frac
1{\alpha_i+\beta_j}\right) \, d(\alpha;\be).
\endgathered
\tag 6.15
$$
\endproclaim
\example{Remark 6.9} The formula (6.15) gives an expression for the integral
$$
\int_{\Delta_{d,d}}s_{\mu}(\alpha;\be)\,
s_{\lambda}(\alpha;\be)\,{\det}^2\left(\frac
1{\alpha_i+\beta_j}\right) \, d(\alpha;\be).
\tag 6.16
$$
The integrand is symmetric in $\la$ and $\mu$. However, our assumptions on $\la$ and $\mu$ are different: the depth of $\la$ must be equal to $d$ while $\mu$ is arbitrary. Actually, if the depth of $\mu$ is $>d$ then both sides of (6.15) vanish. If the depth of $\mu$ is equal to $d$, the integration can be carried out directly in a rather simple way using the Berele--Regev formula
$$
s_\nu(\al_1,\dots,\al_d;\,\be_1,\dots,\be_d)={\left[\det\left(\frac 1{\al_i+\be_j}\right)_{i,j=1}^d\right]}^{-1}\,
\det[\al_i^{P_j}]_{i,j=1}^d\,\det[\be_i^{Q_j}]_{i,j=1}^d;
$$
here $(P|\,Q)$ are the Frobenius coordinates of $\nu$, see
\cite{BR}, \cite{M, I.3, Ex.23}. But if the depth of $\mu$ is strictly less than $d$, the Berele--Regev formula must be replaced by the more complicated Sergeev--Pragacz formula, and a direct integration seems to be more difficult.
\endexample
\example{Remark 6.10} All claims of this subsection remain true if
we replace integral Frobenius coordinates $p_1>\dots>p_d\ge 0$,
$q_1>\dots>q_d\ge 0$ of a fixed Young diagram $\la$ with
any ordered sequences of real numbers $>-\frac 12$. Then the Schur
function $s_\la(\al;\be)$ should be understood as
$$
{\left[\det\left(\frac 1{\al_i+\be_j}\right)_{i,j=1}^d\right]}^{-1}\,
\det[\al_i^{p_j}]_{i,j=1}^d\,\det[\be_i^{q_j}]_{i,j=1}^d,
$$
cf. Remark 6.5. As in 6.1, we restricted ourselves to integral $p$'s and $q$'s in order to demonstrate the symmetry $\la\leftrightarrow\mu$ in (6.15). By analytic continuation, the formula (6.15) can be extrapolated to any pairwise distinct complex $p_i$'s and $q_i$'s.
\endexample
\subhead 6.3. Truncated Kingman branching \endsubhead
For the Kingman graph $\K$ (see \S4) the algebra $A^*$ coincides with
the algebra $\Lambda$ of symmetric functions.
We consider specializations of $\Lambda$ obtained by fixing finitely
many indeterminates, say, $x_1,\dots,x_l$, and sending remaining indeterminates to zero.
Under such a specialization all functions $\P_\mu^*=m_\mu^*$ with $\ell(\mu)>l$ vanish (this easily follows from the definition of the factorial monomial functions, see \S4). This means that the corresponding harmonic function lives on the subgraph $\K(l)$ of the Kingman graph $K$ consisting of all Young diagrams with the length $\le l$. We call this subgraph the {\it truncated Kingman graph}. It fits into the general formalism of \S1 with algebras $A$ and $A^*$ both equal to the algebra of symmetric polynomials in $l$ variables. The elements $\P_\mu$ and $\P_\mu^*$ are monomial symmetric polynomials and factorial monomial symmetric polynomials, respectively.
Certain harmonic functions on truncated Kingman graphs and their applications to Selberg--type integrals were previously considered by Kerov \cite{Ke3}.
Let us fix a Young diagram
$$
\la=(\la_1\ge\dots\ge\la_l>0)=(1^{r_1(\la)}2^{r_2(\la)}\dots).
$$
We define an algebra homomorphism $\pi_\la:\La\to \R$ as follows, cf.
(6.1), (6.11),
$$
\cases
x_i\mapsto-\la_i-1, &1\le i\le l,\\
x_i\mapsto 0, &i>l.
\endcases
\tag 6.17
$$
The corresponding harmonic function on $\K$ has the form
$$
\varphi_{\lambda}(\mu)=\frac{(-1)^{|\mu|}\pi_\la(m^*_\mu)}{(-\pi_\la(m^*_{(1)}))_{|\mu|}}=
\frac{(-1)^{|\mu|}m^*_\mu(-\lambda_1-1,\dots,-\la_l-1)}{(|\lambda|+l)_{|\mu|}}.
\tag 6.18
$$
\proclaim{Proposition 6.11} The harmonic function $\varphi_\la$ defined by \tht{6.18} is nonnegative.
\endproclaim
\demo{Proof}
It suffices to note that
$m^*_\mu(-\lambda_1-1,\dots,-\la_l-1)$, by definition, is a sum of expressions of the
form ($\sigma$ is a permutation here)
$$
\prod_{i=1}^{l(\mu)}(-\la_{\sigma(i)}-1)(-\la_{\sigma(i)}-2)
\cdots(-\la_{\sigma(i)}-\mu_i),
$$
each of which has sign $(-1)^{|\mu|}$.\qed
\enddemo
According to \S1, we have probability distributions $M_n^\la$ on $\K_n$
for all $n=1,2,\ldots$ given by
$$
M_n^{\la}(\nu)=\dim_\K(\nu)\,\varphi_{\la}(\nu), \quad |\nu|=n.
\tag 6.19
$$
Embeddings $\K_n\hookrightarrow\Om(\K)$ are defined as follows
($\Om(\K)$ was defined in (4.7)):
$$
\nu\in\K_n\mapsto\left(\frac{\nu_1}n,\frac{\nu_2}n,\dots\right)\in\Om(\K).
\tag 6.20
$$
The images of the probability distributions $M_n^{\la}$ via these
embeddings, according to the general theory, weakly converge, as $n\to\infty$, to a certain
probability measure $P$ on $\Om(\K)$.
\proclaim{Proposition 6.12} The probability measure $P$ on
$\Om(\K)$ is supported by the finite--dimensional face
$$
\gather
\Delta_l=\{\alpha\in\Om(\K)\,|\,\al_{l+1}=\al_{l+2}=\dots=0,\
\sum_{i=1}^l\al_i=1\}\\ \backsimeq
\{(\al_1,\ldots,\al_l)\in \Bbb R^l_+|\,\al_1\ge
\al_2\ge\dots\ge \al_l,\ \sum_{i=1}^l\al_i=1 \}.
\endgather
$$
Its density with respect to the Lebesgue measure $d\al$ on $\Delta_l$ equals
$$
\frac{\Gamma(|\la|+l)
\cdot r_1(\la)!r_2(\la)!\cdots}{\prod_{i=1}^l\la_i!}\,
m_\la(\al_1,\dots,\al_l).
\tag 6.21
$$
\endproclaim
The proof is very similar to that of Proposition 6.2, so we shall
just state the analog of Lemma 6.3 in this case.
\proclaim{Lemma 6.13} As $n\to\infty$,
$$
\max_{\nu\in\K_n} M_n^\la(\nu)=O(n^{1-l}).
$$
Furthermore, if $\nu_{l+1}=0$ and $\nu_l\ge \varepsilon n$, where $\varepsilon>0$ is arbitrary, then, as $n\to\infty$,
$$
M_n^{\la}(\nu)=n^{1-l}\, \frac{\Gamma(|\la|+l)
\cdot r_1(\la)!r_2(\la)!\cdots}
{\prod_{i=1}^l\la_i!}\,m_\la(\al_1,\dots,\al_l)\,(1+o(1)),
\tag 6.22
$$
with $
\al_1={\nu_1}/n,\dots,\al_l={\nu_l}/n.
$
The estimate \tht{6.22} is uniform in $\nu$.
\endproclaim
\demo{Proof} Using the formula (4.6) for $\dim_\K(\nu)$ we get
$$
M_n^{\la}(\nu)=\dim_\K(\nu)\,\varphi_\la(\nu)=
\frac{n!}{(|\la|+l)_{n}}\,\frac {\sum_{\sigma}
\prod_{i=1}^{\ell(\nu)}(\la_{\sigma(i)}+1)_{\nu_i}}{\nu_1!\cdots\nu_{\ell(\nu)}!},
$$
where the summation is taken over all permutations $\sigma\in S_l$ which produce different products $\prod_{i=1}^{\ell(\nu)}(\la_{\sigma(i)}+1)_{\nu_i}$.
Next, we have asymptotic relations
$$
\frac {n!}{(|\la|+l)_n}=O(n^{1-l-|\la|}),
$$
$$
\frac{(\la_j+1)_{\nu_i}}{\nu_i!}\le (\la_j+\nu_i)^{\la_j}=O(n^{\la_j}).
$$
They imply that each term of the sum above, after multiplication by remaining factors, is at most of order
$n^{1-l}$. This proves the first part of the lemma.
For the second part of the lemma, assume $\nu_l\ge \varepsilon n$. Then we have
$$
\frac {n!}{(|\la|+l)_n}=\Gamma(|\la|+l)\, n^{1-l-|\la|}(1+o(1)),
$$
$$
\frac{(\la_j+1)_{\nu_i}}{\nu_i!}=\frac {(\al_in)^{\la_j}}{\la_j!}\,(1+o(1))
$$
as $n\to\infty$, all estimates are uniform in $\nu\in \K_n$ provided that $\nu_l\ge \varepsilon n$. This yields the estimate (6.22).\qed
\enddemo
As a corollary, we get the Poisson integral representation, cf.
Propositions 6.4 and 6.8.
\proclaim{Proposition 6.14}
$$
\gathered
\frac{(-1)^{|\mu|}m^*_\mu(-\lambda_1-1,\dots,-\la_l-1)}{(|\lambda|+l)_{|\mu|}}\\=
\frac{\Gamma(|\la|+l)\cdot r_1(\la)!r_2(\la)!\cdots}{\prod_{i=1}^l
\la_i!}\int_{\Delta_l}m_\mu(\al)m_\la(\al)d\al.
\endgathered
\tag 6.23
$$
\endproclaim
This claim, similarly to Proposition 6.4, can be proved directly by making use of Dirichlet integrals.
\example{Remark 6.15} All claims above remain valid for any positive ordered sequence $\la=(\la_1\ge\la_2\ge\dots\ge\la_l\ge 0)$ with the
obvious modification of the definition of monomial symmetric function $m_\la(\al)$, cf. Remarks 6.5, 6.10. By analytic continuation, the formula (6.23) can be extended to arbitrary complex $\la_i$'s.
\endexample
\subhead 6.4. Truncated Schur branching \endsubhead
In this subsection we shall deal with the Schur graph, see \S5. For this graph the algebras $A$ and $A^*$ coincide with a subalgebra $\Gamma$ of the algebra of symmetric functions; $\Gamma$ is generated by the odd power sums $\sum_ix_i^{2k+1}$, $k=0,1,\dots$ . Again, we consider specializations of $\Gamma$ obtained by fixing variables $x_1,\dots,x_l$ and sending remaining variables to zero. As in 6.2, we shall state the results and sketch the ideas of the proofs.
In such a specialization, the elements $\P_\mu^*=P_\mu^*$ vanish if $\ell(\mu)>l$. This means that the corresponding harmonic function can be viewed as a harmonic function on the {\it truncated Schur graph} $\SS(l)$ --- the subgraph of the Schur graph $\SS$ consisting of diagrams with length $\le l$. The truncated Schur graphs also fit into the formalism of \S1 with algebras $A$ and $A^*$ coinciding with the subalgebra of the algebra of symmetric polynomials in $l$ variables generated by odd power sums. The elements $\P_\mu$ and $\P_\mu^*$ are the Schur $P$ polynomials and the factorial Schur $P$ polynomials, respectively.
Let us fix a strict partition $\lambda$ and denote its length by $l$.
We define a multiplicative linear functional
$\pi_\la:\Gamma\to \R$ as follows, cf. (6.1), (6.11), (6.17):
$$
\cases
x_i\mapsto-\la_i-1, &1\le i\le l,\\
x_i\mapsto 0, &i>l.
\endcases
$$
The corresponding harmonic function on $\SS$ has the form
$$
\varphi_{\lambda}(\mu)=\frac{(-1)^{|\mu|}\pi_\la(P^*_\mu)}{(-\pi_\la(P^*_{(1)}))_{|\mu|}}=\frac{(-1)^{|\mu|}P^*_\mu(-\lambda_1-1,\dots,-\la_l-1)}
{(|\lambda|+l)_{|\mu|}}.
\tag 6.24
$$
As in 6.2, it is not evident that this function is nonnegative.
For all strict partitions $\nu\in\SS_n$, $n=1,2,\ldots$, we define, as usual,
$$
M_n^{\la}(\nu)=\dim_\SS\nu\cdot\varphi_\la(\nu).
$$
Embeddings $\SS_n \hookrightarrow\Om(\SS)=\Om(\K)$ are defined
exactly as in the case of the Kingman graph, see (6.20), the only
difference is that now all partitions are strict.
\proclaim{Proposition 6.16} The sequence of images of (possibly signed) measures $M_n^{\la}$ under the embeddings defined above weakly
converges, as $n\to\infty$, to a (positive) probability measure $P$ on $\Om(\SS)$. This
measure is supported by the finite--dimensional face
$$
\gather
\Delta_l=\{\alpha\in\Om(\SS)\,|\,\al_{l+1}=\al_{l+2}=\dots=0,\
\sum_{i=1}^l\al_i=1\}\\ \backsimeq
\{(\al_1,\ldots,\al_l)\in \Bbb R^l_+|\,\al_1\ge
\al_2\ge\dots\ge \al_l,\ \sum_{i=1}^l\al_i=1 \},
\endgather
$$
and its density with respect to the Lebesgue measure $d\al$ on $\Delta_l$ equals
$$
\frac{\Gamma(|\lambda|+l)}{\prod_{i=1}^l\lambda_i!}
\left[\pf\left(\frac{\lambda_i-\lambda_j}{\lambda_i+\lambda_j+2}\right)
\right]^{-1}P_\lambda(\alpha_1,\dots,\al_l)\,
{\pf}^2\left(\frac{\alpha_i-\alpha_j}{\alpha_i+\alpha_j}\right).
\tag 6.25
$$
\endproclaim
This result is parallel to Propositions 6.2, 6.7, 6.12. Its proof is based on an appropriate analog of the approximation Lemmas 6.3, 6.13. The proof of such a lemma in this case follows from explicit formulas for Schur $P$--functions and factorial Schur functions \cite{M, III.8}, \cite{I1}.
Similarly to Proposition 6.7, Proposition 6.16 implies the existence of the Poisson integral representation for $\varphi_\la$. Thanks to the positivity of (6.25), this implies that $\varphi_\la$ is nonnegative.
The explicit Poisson integral representation (1.2) in this case takes the following form.
\proclaim{Proposition 6.17}
$$
\gathered
\frac{(-1)^{|\mu|}P^*_\mu(-\lambda_1-1,\dots,-\la_l-1)}{(|\lambda|+l)_{|\mu|}}=\frac{\Gamma(|\lambda|+l)}
{\prod_{i=1}^l\lambda_i!}
\left[\pf\left(\frac{\lambda_i-\lambda_j}{\lambda_i+\lambda_j+2}\right)
\right]^{-1}\\ \times
\int\limits_{\Delta_l}P_\mu(\al)P_\la(\al)\pf^2\left(\frac{\al_i-\al_j}
{\al_i+\al_j}\right)\,d\al.
\endgathered
\tag 6.26
$$
\endproclaim
\example{Remark 6.18} If $\ell(\mu)=l\;(=\ell(\la))$, then the integration in (6.26) can be carried out directly using Dirichlet integrals and the formula \cite{M, III.8, Ex.12}
$$
P_\nu(\al_1,\dots,\al_l)=\left[\pf\left(\frac{\al_i-\al_j}{\al_i+\al_j}\right)\right]^{-1}
\det[\al_i^{\nu_j}]_{i,j=1}^l\,,
\tag 6.27
$$
which holds for all $\nu$ of length $l$. If $\ell(\mu)\la_2>\dots>\la_l\ge 0)$. Then the Schur $P$--function
$P_\la(\al)$ should be understood as
$$
\left[\pf\left(\frac{\al_i-\al_j}{\al_i+\al_j}\right)\right]^{-1}
\det[\al_i^{\la_j}]\,,
$$
cf. Remarks
6.5, 6.10, 6.15.
By analytic continuation, the formula (6.26) can be extended to arbitrary complex mutually distinct $\la_i$'s.
\endexample
\head \S7. Appendix \endhead
\demo{Proof of Theorem 1.1} Existence of the integral representation
\tht{1.2} follows from Choquet's theorem, see, e.g., \cite{Ph}. To
prove its uniqueness one can apply another theorem, due to Choquet
and Meyer, \cite{DM}. Then we have to verify that the cone
$\Harm^+(\G)$ is a lattice, i.e., for any
$\varphi,\psi\in\Harm^+(\G)$, there exist their lowest upper bound
$\varphi\vee\psi$ and greatest lower bound $\varphi\wedge\psi$. Let
us prove that
$$
\gather
(\varphi\vee\psi)=\lim_{n\to\infty}\sum_{\la\in\G_n}
\dimG(\mu,\la)\max(\varphi(\la), \psi(\la)), \tag7.1 \\
(\varphi\wedge\psi)=\lim_{n\to\infty}\sum_{\la\in\G_n}
\dimG(\mu,\la)\min(\varphi(\la), \psi(\la)). \tag7.2 \\
\endgather
$$
Indeed, take \tht{7.1} and \tht{7.2} as the definition of the
functions $\varphi\vee\psi$ and $\varphi\wedge\psi$. For any fixed
$\mu$, the sum in the right--hand side of \tht{7.1}
increases as $n\to\infty$ and remains bounded from above by
$\varphi(\mu)+\psi(\mu)$. Similarly, the sum in the right--hand side of \tht{7.2}
decreases and remains bounded from below by 0. Hence, the limits
exist.
Next, remark that for any fixed vertex $\nu$, the function
$\mu\mapsto\dimG(\mu,\nu)$ satisfies the harmonicity relation up to
level $|\nu|-1$. This implies that $\varphi\vee\psi$ and
$\varphi\wedge\psi$ are harmonic functions. Clearly, they are
nonnegative and are upper and lower bounds, respectively.
Finally, it is readily verified that they are the lowest upper bound and
the greatest lower bound, respectively. \qed
\enddemo
\newpage
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\enddocument
\bye