# The Schur–Weyl graph and Thoma’s theorem††thanks: Supported by the RSF grant
21-11-00152.

A. M. Vershik St. Petersburg Department of Steklov Institute of Mathematics,
St. Petersburg State University, Institute for Information Transmission
Problems. E-mail<EMAIL_ADDRESS>N. V. Tsilevich St. Petersburg
Department of Steklov Institute of Mathematics. E-mail<EMAIL_ADDRESS>
###### Abstract

We define a graded graph, called the Schur–Weyl graph, which arises naturally
when one considers simultaneously the RSK algorithm and the classical duality
between representations of the symmetric and general linear groups. As one of
the first applications of this graph, we give a new proof of the completeness
of the list of discrete indecomposable characters of the infinite symmetric
group.

Key words: Schur–Weyl graph, RSK algorithm, Thoma theorem, central measures.

To the memory of S. V. Kerov (12.06.1946–30.07.2000)

##  1 Introduction

In [8], it was shown that the RSK algorithm generalized to infinite
(Bernoulli) sequences allows one to enumerate all characters of the infinite
symmetric group. However, this observation did not provide a new proof of the
completeness of the list of indecomposable characters (i.e., of Thoma’s
theorem [14]), but it revealed some profound properties of measures
corresponding to characters. The so-called youngization homomorphism from the
$L^{2}$ Bernoulli space onto the Hilbert space of functions on infinite Young
tableaux endowed with the resulting central measure, first, is an isomorphism,
as is proved in the very important papers [11, 12], and, second, allows one to
give new realizations of some irreducible representations of the infinite
symmetric group.

The finite RSK algorithm associates with a finite word in an ordered alphabet
a pair of Young tableaux, a semistandard $P$-tableau and a standard
$Q$-tableau. It is in this form that it is considered in most of the
literature. The key element of our approach is a “dynamic” view (first
suggested in [8]) on the RSK algorithm, when it is applied to growing
sequences of symbols and one considers the dynamics of the arising tableaux.
The youngization homomorphism from [8] uses the $Q$-tableau, which also just
grows. However, the behavior of the $P$-tableau is much more complicated.
Nevertheless, as this paper shows (see also [17]), tracking the dynamics of
the $P$-tableau is also doable and very useful. It is in this way that we have
led to the definition of the Schur–Weyl graph, which is a covering of the
Young graph and reflects the dynamics of the $P$-tableau when the RSK
algorithm is applied to a growing sequence of symbols. Besides, this graph is
related in a natural way to the Schur–Weyl duality between representations of
the general linear groups and symmetric groups. The relation between the RSK
algorithm and the Schur–Weyl duality was observed earlier, but the authors
have not been able to find precise statements in the literature. It is clear
that this duality must show itself also in the infinite version of the
algorithm; further research is especially needed in this direction. We
emphasize that in the general context of this paper, an important role is
played by the machinery of the theory of filtrations (decreasing sequences of
measurable partitions), which was developed for the needs of ergodic theory,
but turned out to be adequate for the problems under consideration (see, e.g.,
[21]).

In this paper, we use the Schur–Weyl graph to obtain a new proof of Thoma’s
theorem [14] in the case of discrete central measures. There are several
proofs of this important theorem ([20], [10], [7], etc., see the survey [16]);
however, our argument is much simpler and of a purely combinatorial nature.
Like many other proofs, it relies on the so-called ergodic method [15], but
the direct application of this method to the Young graph entails the need for
calculating the asymptotics of dimensions of skew Young diagrams; this is a
difficult problem, which makes it necessary to develop special techniques,
such as, e.g., shifted Schur functions. However, using the RSK algorithm and
the Schur–Weyl graph allows us to reduce the problem under consideration to
calculating the asymptotics of dimensions of ordinary Young diagrams, which is
a much easier task. For another combinatorial argument for the same discrete
case, see [19]. However, it is important to mention that the difficulty of the
problem of describing the central measures is significantly different in the
discrete and continuous (zero frequencies) cases.

The paper is organized as follows. In Section 2, we consider the youngization
homomorphism from the space of infinite words to the space of infinite Young
tableaux and its properties. In Section 3, the Schur–Weyl graph is introduced
and it is shown how the problem of describing the central measures on the
Young graph can be reduced to the analogous problem for the Schur–Weyl graph.
In Section 4, we apply the ergodic method to solve this problem for the
Schur–Weyl graph in the case of finitely many rows; the proof is split into
lemmas, presented in Sections 4.1–4.3, and put together in Section 4.4.
Finally, in Section 5 we explain how the obtained result can be extended to
the case of central measures supported on tableaux with finitely many rows and
columns.

The reader is assumed to be familiar with the Robinson–Schensted–Knuth (RSK)
algorithm and its basic properties (see, e.g., [13, 4]); for background on the
representation theory of the infinite symmetric group (the Young graph,
central measures, Thoma parameters, etc.), see, e.g., [6, 3].

Throughout the paper, Young tableaux are filled with symbols from a fixed
alphabet ${\cal A}$.

##  2 Youngization

Let $k\in{\mathbb{N}}$ and consider the alphabet ${\cal A}=\\{1,2,\ldots,k\\}$
consisting of $k$ ordered symbols. Consider the space $X={\cal A}^{\infty}$ of
infinite words in the alphabet ${\cal A}$.

By $\operatorname{RSK}(w)=(P(w),Q(w))$ we denote the result of applying the
Robinson–Schensted–Knuth (RSK) correspondence to a finite sequence (word) $w$
in the alphabet ${\cal A}$; thus, $P(w),Q(w)$ is a pair of Young tableaux of
the same shape (with at most $k$ rows), which will be denoted by
$\operatorname{sh}(w)$; the tableau $P(w)$ is semistandard, while the tableau
$Q(w)$ is standard. Given an infinite sequence $x\in X$, denote by
$[x]_{n}\in{\cal A}^{n}$ its initial segment of length $n$.

In [8], the following map from the space of infinite sequences to the space
${\cal T}({\mathbb{Y}})$ of infinite standard Young tableaux (or, which is the
same, the set of infinite paths in the Young graph ${\mathbb{Y}}$) was
introduced.

######  Definition 1.

Apply the $\operatorname{RSK}$ algorithm to the initial segments $[x]_{n}$ of
a sequence $x\in X$ and set $\operatorname{RSK}([x]_{n})=(P_{n}(x),Q_{n}(x))$.
It is clear from the construction of this algorithm that
$\lim\limits_{n\to\infty}Q_{n}(x)=Q(x)$ is an infinite standard Young tableau;
denote it by $\pi(x)$. The resulting map $\pi:X\to{\cal T}({\mathbb{Y}})$ is
called the youngization.

Endow $X$ with a Bernoulli measure $m_{p}^{\infty}$, where
$p=(p_{1},p_{2},\ldots,p_{k})$, $p_{i}=\operatorname{Prob}(i)$, and let ${\cal
M}_{(p,0,0)}$ be the central measure on ${\cal T}({\mathbb{Y}})$ with Thoma
parameters $(p,0,0)$ (note that the measure ${\cal M}_{(p,0,0)}$ is supported
on the subset of tableaux with at most $k$ rows). In [8], it is proved that
the youngization is a homomorphism of measure spaces

$\pi:(X,m_{p}^{\infty})\to({\cal T}({\mathbb{Y}}),{\cal M}_{(p,0,0)}).$ (1)

The following measurable partitions are defined in a natural way on the space
of infinite Young tableaux ${\cal T}({\mathbb{Y}})$: the cylinder partition
$\xi_{n}$ of level $n$, whose element is the set of infinite paths in the
Young graph with fixed initial segment of length $n$ and arbitrary “tail”, and
the tail partition $\eta_{n}$ of level $n$, whose element is the set of
infinite paths in the Young graph with fixed $(n+1)$-tail and arbitrary
beginning. In [21], we introduced the following partitions on the space $X$.

######  Definition 2.

The Young cylinder partition and the Young tail partition of the space $X$ of
infinite sequences are the partitions $\bar{\xi}_{n}:=\pi^{-1}\xi_{n}$ and
$\bar{\eta}_{n}:=\pi^{-1}\eta_{n}$, respectively.

Clearly, the sequence of partitions $\bar{\xi}_{n}$ is monotonically
increasing, while the sequence of partitions $\bar{\eta}_{n}$ (called the
Young filtration) is monotonically decreasing. Recall (see, e.g., [9]) that
the plactic class ${\cal P}_{t}$ (respectively, the coplactic class
${\mathfrak{C}}_{t}$) corresponding to a given Young tableau $t$ of size $n$
is the set of all words $u$ of length $n$ such that $P(u)=t$ (respectively,
$Q(u)=t)$. In [21], it is shown that the Young partitions on the space $X$ can
be described as follows:

  * •

the elements of $\bar{\xi}_{n}$ are indexed by the standard Young tableaux $t$
of size $t$ and coincide with the coplactic classes ${\mathfrak{C}}_{t}$;

  * •

the elements of $\bar{\eta}_{n}$ are indexed by the pairs $(t,y)$, where $t$
is a semistandard Young tableaux of size $n$ and $y\in X$, and have the form
$\\{x\in X:[x]_{n}\in{\cal P}_{t},\,(x_{n+1},x_{n+2},\ldots)=y\\}$.

Also, in [21] it is proved that

  * •

the sequence of partitions $\bar{\xi}_{n}$ is total, i.e., converges in the
weak topology to the partition $\varepsilon$ into singletons;

  * •

the sequence of partitions $\bar{\eta}_{n}$ is ergodic, i.e., converges in the
weak topology to the trivial partition $\nu$.

In particular, it follows that the youngization (1) is in fact an isomorphism
of measure spaces, which gives (in this special case of finitely many rows)
another proof of this result, originally proved in [12].

It is natural to pose the problem of describing all ergodic measures on the
space $X$ invariant with respect to the Young tail partitions. In the next
section, we introduce the so-called Schur–Weyl graph, for which this problem
is equivalent to the problem of describing all ergodic central measures.

##  3 The Schur–Weyl graph

######  Definition 3.

The Schur–Weyl graph $\operatorname{SW}_{k}$ is the graded graph defined as
follows:

  * •

the vertices of level $n$ are all semistandard Young tableaux with $n$ cells
filled with symbols from the alphabet ${\cal A}$;

  * •

tableaux $t_{1}$ and $t_{2}$ are joined by an edge if $t_{2}$ is obtained from
$t_{1}$ by row inserting an arbitrary element $i\in{\cal A}$.

Clearly, the graph $\operatorname{SW}_{k}$ has the following properties:

  * •

every vertex has exactly $k$ outgoing edges;

  * •

the dimension of a vertex $t$ (the number of paths leading to this vertex from
the initial vertex $\emptyset$ of level $0$) is $\dim\lambda$, where
$\lambda=\operatorname{sh}(t)$ is the shape of $t$.

Note that the graph $\operatorname{SW}_{k}$ can be regarded as a covering of
the Young graph ${\mathbb{Y}}$ (more exactly, of the subgraph
${\mathbb{Y}}_{k}$ in ${\mathbb{Y}}$ consisting of the diagrams with at most
$k$ rows): the fiber over a vertex (Young diagram) $\lambda\in{\mathbb{Y}}$
consists of all semistandard tableaux of shape $\lambda$. On the other hand,
this construction is related in a natural way to the classical Schur–Weyl
duality (see below).

Example. For $k=2$, the Schur–Weyl graph $\operatorname{SW}_{2}$ has the
following form:

  * •

the vertices of level $n$ are all pairs of the form $(\lambda,r)$ where
$\lambda=(\lambda_{1},\lambda_{2})$ is a Young diagram with $n$ cells and at
most two rows and $r$ is an integer from the interval $0\leq
r\leq\lambda_{1}-\lambda_{2}$. Set $k(\lambda):=\lambda_{1}-\lambda_{2}$.

  * •

$(\lambda,r)\nearrow(\mu,s)$ $\iff$ (i) $\lambda\nearrow\mu$ (i.e., $\mu$ is
obtained from $\lambda$ by adding one cell) and (ii) $s=r$ or $s=r+1$ if
$k(\mu)=k(\lambda)+1$, and $s=r$ if $k(\mu)=k(\lambda)-1$.

######  Lemma 1.

The set ${\cal T}_{n}(\operatorname{SW}_{k})$ of all paths of length $n$
(respectively, the space ${\cal T}(\operatorname{SW}_{k})$ of infinite paths)
in the Schur–Weyl graph can be identified in a natural way with the set ${\cal
A}^{n}$ of words of length $n$ (the space $X={\cal A}^{\infty}$ of infinite
words) in the alphabet ${\cal A}$.

###### Proof.

A sequence $u=(x_{1},\ldots,x_{n})\in{\cal A}^{n}$ is identified with the
vertex $t=P(u)$ where $(P(u),Q(u))=\operatorname{RSK}(u)$. ∎

Thus, the space of functions on paths of length $n$ in the graph
$\operatorname{SW}_{k}$ can be identified in a natural way with the space
$(\mathbb{C}^{k})^{\otimes n}$; then the subspace corresponding to the paths
leading to tableaux with a given diagram $\lambda$ (i.e., lying in the fiber
over the vertex ${\lambda\in{\mathbb{Y}}_{k}}$ of the Young graph) gets
identified with the irreducible, with respect to
$\operatorname{GL}(k,\mathbb{C})\otimes{\mathfrak{S}}_{n}$, component
$M_{\lambda}\otimes V_{\lambda}$ of the Schur–Weyl decomposition, where
$V_{\lambda}$ is the space of the irreducible representation of the symmetric
group ${\mathfrak{S}}_{n}$ corresponding to the diagram $\lambda$ and
$M_{\lambda}$ is the space of the irreducible representation of the group
$\operatorname{GL}(k,\mathbb{C})$ corresponding to this diagram. One may say
that ${\mathfrak{S}}_{n}$ acts on the “base”, i.e., on the space spanned by
the paths in the Young graph ${\mathbb{Y}}_{k}$ leading to the given vertex
$\lambda$, while $\operatorname{GL}(k,\mathbb{C})$ acts in the “fibers” of the
graph $\operatorname{SW}_{k}$ over each such vertex.

The description of Young partitions on the space $X$ implies the following
result.

######  Proposition 1.

Under the identification from Lemma 1, the Young tail partition
$\bar{\eta}_{n}$ on the space $X$ is identified with the tail partition of
level $n$ on the space ${\cal T}(\operatorname{SW}_{k})$ of infinite paths in
the Schur–Weyl graph.

######  Corollary 1.

Every (ergodic) central measure on the space ${\cal T}(\operatorname{SW}_{k})$
of infinite paths in the Schur–Weyl graph $\operatorname{SW}_{k}$ is the
image, under the identification from Lemma 1, of an (ergodic) measure on the
space $X$ invariant with respect to the Young filtration.

On the other hand, the following lemma shows that in order to find all central
measures on the Young graph, it suffices to find all central measures on the
Schur–Weyl graph.

######  Lemma 2.

For every (ergodic) central measure $\nu$ on the subgraph ${\mathbb{Y}}_{k}$
of the Young graph there exists an (ergodic) central measure $\mu$ on the
Schur–Weyl graph $\operatorname{SW}_{k}$ that projects to $\nu$ under the
youngization.

###### Proof.

For each Young diagram $\lambda$, consider the “maximal” tableau $S_{\lambda}$
of shape $\lambda$ in which each column of length $i$ is filled with the
consecutive integers $k-i+1,\ldots,k$. Clearly, the subgraph in
$\operatorname{SW}_{k}$ consisting of the maximal tableaux is isomorphic to
${\mathbb{Y}}_{k}$. The image $\mu$ of the measure $\nu$ under this
isomorphism is as required. Obviously, if $\nu$ is ergodic, then $\mu$ is also
ergodic. ∎

To describe the central measures, we use the ergodic method [15], in its
version for central measures on the path space of a graded graph presented in
[6]. Namely, let $\Gamma$ be a graded graph and ${\cal T}(\Gamma)$ be the
space of infinite paths on $\Gamma$. Given vertices $v,w$ in $\Gamma$, denote
by $\dim(v,w)$ the number of paths from $v$ to $w$, and let $\dim
v:=\dim(\emptyset,v)$ be the dimension of $v$. Also, denote by $C_{v}$ the
cylinder set of paths passing through $v$.

######  Theorem 1 (ergodic method [6]).

Let $M$ be an ergodic central measure on the space ${\cal T}(\Gamma)$. Then
for $M$-a.e. path $t=(v_{0},v_{1},\ldots,v_{n},\ldots)$, for all vertices
$v\in\Gamma$, the following limits exist:

$\lim_{n\to\infty}\frac{\dim(v,v_{n})}{\dim v_{n}}=\phi(v).$

The cylinder distributions of the measure $M$ are given by the formula

$M(C_{v})=\dim v\cdot\phi(v).$

In other words, let ${\mathfrak{C}}_{n}(t)$ be the class of the tail partition
of level $n$ containing the finite segment $t_{n}=(v_{0},\ldots,v_{n})$ of an
infinite path $t$ and $\mu_{t}^{(n)}$ be the uniform measure on
${\mathfrak{C}}_{n}(t)$; then for $M$-a.e. path $t$ the measures
$\mu_{t}^{(n)}$ weakly converge to $M$.

##  4 Realization of the ergodic method

Our purpose is to use the Schur–Weyl graph to prove Thoma’s theorem, which
gives the list of all ergodic central measures on the Young graph, in the case
of measures supported on tableaux with finitely many rows and finitely many
columns. To make the proof clearer, in this section we consider the case of
finitely many rows, and then in Section 5 describe how the result can be
extended to a more general case.

######  Theorem 2.

All ergodic central measures on the subgraph ${\mathbb{Y}}_{k}$ of the Young
graph consisting of the diagrams with at most $k$ rows are exhausted by the
Thoma measures ${\cal M}_{(p,0,0)}$ where $p=(p_{1},\ldots,p_{k})$,
$p_{1}\geq\ldots\geq p_{k}\geq 0$, $p_{1}+\ldots+p_{k}=1$.

For convenience, the first two subsections of this section contain the lemmas
used in the proof: in Section 4.1 it is shown that, given a sequence $x\in X$,
the normalized row lengths of the growing diagram $\operatorname{sh}([x]_{n})$
coincide with the limiting densities of elements in $x$; Section 4.2 contains
combinatorial lemmas related to the properties of the RSK algorithm; and
Section 4.3 contains lemmas on the asymtpotics of dimensions of Young diagrams
and on “convolution” of determinants.

###  4.1 Densities of elements and normalized row lengths

######  Proposition 2.

Given $x\in X$, let
$\operatorname{sh}([x]_{n})=(\lambda_{1}^{(n)},\ldots,\lambda_{k}^{(n)})$. If
the limiting densities of elements

$p_{j}:=\lim_{n\to\infty}\frac{\\#\\{i\leq n:x_{i}=j\\}}{n},\quad
j=1,\ldots,k,$ (2)

exist, then the normalized row lengths

$\lim_{n\to\infty}\frac{\lambda_{j}^{(n)}}{n}=\hat{p}_{j},\quad j=1,\ldots,k,$
(3)

exist too, where $\hat{p}_{1},\ldots,\hat{p}_{k}$ are the densities
$p_{1},\ldots,p_{k}$ arranged in nonincreasing order:
$\hat{p}_{1}\geq\ldots\geq\hat{p}_{k}$.

###### Proof.

Induction on the size $k$ of the alphabet ${\cal A}$.

######  Lemma 3.

Proposition 2 holds for the alphabet ${\cal A}_{2}=\\{1,2\\}$.

###### Proof.

It is convenient to regard a sequence from ${\cal A}_{2}^{n}$ as a word
$w=x_{1}\ldots x_{n}$ in the alphabet ${\cal A}_{2}$. Bracket every factor
$21$ in $w$. The letters that are not bracketed form a subword $w_{1}$ in $w$.
Bracket every factor $21$ in $w_{1}$. There remains a subword $w_{2}$.
Continue this procedure until we are left with a word of the form
$w_{k}=1^{a}2^{b}=x_{i_{1}}\ldots x_{i_{a+b}}$ with $a,b\geq 0$. We say that
the coordinates $x_{i_{1}},\ldots,x_{i_{a+b}}$ of $w$ are free and the other
coordinates are paired. Then $\lambda_{2}^{(n)}$ is the number of pairs of
paired coordinates in $[x]_{n}$.

Note that if $p_{1}<p_{2}$, then for sufficiently large $n$ all $1$s in
$[x]_{n}$ are paired, hence the second row grows with the same rate as the
number of $1$s, so the lemma in this case is obvious. Below we assume that
$p_{1}\geq p_{2}$ (in particular, $p_{2}\leq 1/2$).

Let $a_{n}:=\\#\\{i\leq n:x_{i}=1\\}$, $b_{n}:=\\#\\{i\leq n:x_{i}=2\\}$ and
assume that $\lim\limits_{n\to\infty}\frac{b_{n}}{n}=p_{2}$ (then, obviously,
$\lim\limits_{n\to\infty}\frac{a_{n}}{n}=p_{1}$). Setting
$r_{n}=\lambda_{2}^{(n)}$, we will prove that
$\lim\limits_{n\to\infty}\frac{r_{n}}{n}=p_{2}$. Assume to the contrary that
(since, obviously, ${r_{n}\leq b_{n}}$) there exists a subsequence
$n_{i}\to\infty$ such that ${r_{n_{i}}<(p_{2}-\delta)n_{i}}$ with $\delta>0$.
On the other hand, for sufficiently large $i$ we have
${|a_{n_{i}}-p_{1}n_{i}|<t\delta n_{i}}$ and $|b_{n_{i}}-p_{2}n_{i}|<t\delta
n_{i}$ by the definition of limit, where the coefficient ${t\in(0,1)}$ is to
be chosen later. Hence, the initial segment $[x]_{n_{i}}$ contains
$a_{n_{i}}-r_{n_{i}}>(p_{1}-p_{2}+(1-t)\delta)n_{i}$ free $1$s and
${b_{n_{i}}-r_{n_{i}}>(1-t)\delta n_{i}}$ free $2$s, with all free $2$s being
to the right of all free $1$s. Pick $m_{i}$ such that among the elements of
$[x]_{n_{i}}$ with indices larger than $m_{i}$ there are $(1-t)\delta n_{i}$
free $2$s (and no free $1$s). Then $m_{i}=n_{i}-(1-t)\delta n_{i}-2k_{i}$
where $k_{i}$ is the number of pairs of paired coordinates with indices larger
than $m_{i}$. Note that $m_{i}>(p_{1}-p_{2}+(1-t)\delta)n_{i}\to\infty$. Then

$\frac{b_{m_{i}}}{m_{i}}=\frac{b_{n_{i}}-(1-t)\delta
n_{i}-k_{i}}{n_{i}-(1-t)\delta
n_{i}-2k_{i}}=\frac{\frac{b_{n_{i}}}{n_{i}}-(1-t)\delta-\frac{k_{i}}{n_{i}}}{1-(1-t)\delta-2\frac{k_{i}}{n_{i}}}<\frac{p_{2}-(1-2t)\delta-u}{1-(1-t)\delta-2u},$

where $u=\frac{k_{i}}{n_{i}}$. For sufficiently small $t$, the right-hand side
decreases in $u$, so it does not exceed the value at $u=0$, i.e.,
$\frac{p_{2}-(1-2t)\delta}{1-(1-t)\delta}<p_{2}-\varepsilon$, where
$\varepsilon>0$ (for sufficiently small $t$). Choosing such $t$, for
sufficiently large $i$ we have $\frac{b_{m_{i}}}{m_{i}}\leq
p_{2}-\varepsilon$, contradicting the fact that
$\lim\limits_{n\to\infty}\frac{b_{n}}{n}=p_{2}$. ∎

Thus, the induction base $k=2$ is proved. Now we prove the induction step from
$k$ to $k+1$. Let $c_{n}:=\\#\\{i\leq n:x_{i}=k+1\\}$. Note that symbols $k+1$
do not affect the growth of the subtableau in $P([x]_{n})$ consisting of
$1,\ldots,k$. Hence, by the induction hypothesis, the row lengths
$\mu_{1}^{(n)},\ldots,\mu_{k}^{(n)}$ of this subtableau satisfy the relations
$\lim\limits_{n\to\infty}\frac{\mu_{j}^{(n)}}{n-c_{n}}=\frac{q_{j}}{1-p_{k+1}}$
and, consequently, $\lim\limits_{n\to\infty}\frac{\mu_{j}^{(n)}}{n}=q_{j}$,
where $(q_{1},\ldots,q_{k})$ is the sequence $(p_{1},\ldots,p_{k})$ arranged
in nonincreasing order. Now, it follows from Lemma 3 that
$\lim\limits_{n\to\infty}\frac{\lambda_{1}^{(n)}}{n}=\max(q_{1},p_{k+1})=\hat{p}_{1}$
and the density of the elements $k+1$ bumped into the second row is
$\min(q_{1},p_{k+1})$. Analogously,
$\lim\limits_{n\to\infty}\frac{\lambda_{2}^{(n)}}{n}=\max(q_{2},\min(q_{1},p_{k+1}))=\hat{p}_{2}$,
the density of the elements $k+1$ bumped into the third row is
$\min(q_{1},q_{2},p_{k+1})$, and so on up to
$\lim\limits_{n\to\infty}\frac{\lambda_{2}^{(k+1)}}{n}=\min(q_{1},\ldots,q_{k},p_{k+1})=\hat{p}_{k+1}$.
∎

###  4.2 Combinatorial lemmas

The dual Robinson–Schensted–Knuth algorithm RSK* differs from the ordinary one
in that an element $i$ bumps the leftmost element greater or equal to $i$. Let
$\operatorname{RSK}^{*}(w)=(P^{*}(w),Q^{*}(w))$. Here $Q^{*}(w)$ is still a
standard tableau, while $P^{*}(w)$ is a dual semistandard tableau, i.e., a
tableau obtained by transposing a semistandard one.

Let $w=w_{1}\ldots w_{n}\in{\cal A}^{n}$. Denote by $\operatorname{rev}(w)$
the word obtained by reversing the order of letters in $w$:
$\operatorname{rev}(w)=w_{n}\ldots w_{1}$. The following relation between RSK
and RSK* is well known (see, e.g., [1]):

$\operatorname{RSK}(w)=(P,Q)\iff\operatorname{RSK}^{*}(\operatorname{rev}(w))=(P^{t},\operatorname{evac}(Q^{t})),$
(4)

where t stands for transpose and $\operatorname{evac}$ is the evacuation
(Schützenberger involution).

It is also well known that the row insertion procedures underlying the
$\operatorname{RSK}$ and $\operatorname{RSK}^{*}$ algorithms are invertible in
the following sense. Let $w=w_{1}\ldots w_{n}\in{\cal A}^{n}$ and
$w^{\prime}=w_{1}\ldots w_{n-1}$. If $P(w)=T$ and we know the cell added at
the last step of the algorithm (i.e., the diagram
$\operatorname{sh}(w^{\prime})$), then a simple combinatorial procedure allows
us to recover $P(w^{\prime})$ and $w_{n}$. Let $\square$ be a corner cell of
$T$. Denote by $\tau(T,\square)$ and $\varepsilon(T,\square)$ (respectively,
$\tau^{*}(T,\square)$ and $\varepsilon^{*}(T,\square)$) the tableau and the
element obtained in this way under the assumption that this cell has been
added at the last step of the $\operatorname{RSK}$ (respectively,
$\operatorname{RSK}^{*}$) algorithm.

Let $T$ be a semistandard tableau, $|T|=n$, $\operatorname{sh}(T)=\nu$. For
$m\leq n$, denote by ${\cal S}_{m}(\nu)$ the set of tableaux of shape
$\nu\setminus\mu$, where $|\mu|=m$, whose cells are filled with integers
$1,\ldots,m$ so that they decrease along each row and each column. Given
$S\in{\cal S}_{m}(\nu)$, we define a word $u=\Phi_{T}(S)\in{\cal A}^{m}$ as
follows. Set ${T_{0}:=T}$ and then, by induction,

$u_{j}:=\varepsilon^{*}(T_{j-1}^{t},\square_{j}^{t}),\qquad
T_{j}:=(\tau^{*}(T_{j-1}^{t},\square_{j}^{t}))^{t},\qquad j=1,\ldots,m,$

where $\square_{j}$ is the cell of $S$ containing $j$ and $\square^{t}_{j}$ is
the corresponding cell of $S^{t}$. Also, denote by $d(S)$ the dimension of the
Young diagram obtained from $\nu$ by deleting all cells of $S$.

######  Lemma 4.

Given $m\leq n$ and $a\in{\cal A}^{m}$, set $c_{a}(T):=\\#\\{y\in{\cal
A}^{n}:{P(y)=T},\,\break[y]_{m}=a\\}$. Then

$c_{a}(T)=\sum_{S\in{\cal S}_{m}(\nu)\colon\Phi_{T}(S)=a}d(S).$

###### Proof.

Given a word $w\in{\cal A}^{n}$, denote by $[w]^{m}$ its final segment of
length $m$. By (4) we have

$c_{a}(T)=\\#\\{z\in{\cal
A}^{n}:P^{*}(z)=T^{t},[z]^{m}=\operatorname{rev}(a)\\},$

which implies the lemma. ∎

Given $u\in{\cal A}^{n}$, we define a word $\Psi(u):=u^{\prime}_{1}\ldots
u^{\prime}_{n}$ as follows: $u^{\prime}_{i}$ is the number of the column to
which a cell has been added at the $i$th step of the algorithm
$\operatorname{RSK}^{*}(u)$.

######  Lemma 5.

$Q(\operatorname{rev}(\Psi(u)))=Q(\operatorname{rev}(u))$. In particular,
$\operatorname{sh}(\operatorname{rev}([w]^{m}))=\operatorname{sh}(\operatorname{rev}([u]^{m}))$
for every $m\leq n$.

###### Proof.

Clearly, $\Psi(u)$ is a lattice word (i.e., every prefix of $\Psi(u)$ contains
at least as many $1$s as $2$s, at least as many $2$s as $3$s, etc.).
Therefore, when calculating $Q^{*}(\Psi(u))$, if the current element is $j$,
then a cell is added to the $j$th column, which means that
$Q^{*}(\Psi(u))=Q^{*}(u)$. By (4), it follows that
$Q(\operatorname{rev}(\Psi(u)))=Q(\operatorname{rev}(u))$. ∎

Given $S\in{\cal S}_{m}(\nu)$, let $w_{S}=w_{1}\ldots w_{m}\in{\cal A}^{m}$
where $w_{i}$ is the number of the row of $S$ containing $i$.

######  Corollary 2.

If $\Phi_{T}(S)=a$, then $\operatorname{sh}(w_{S})=\operatorname{sh}(a)$.

###### Proof.

Note that $\Phi_{T}(S)=a\iff T=P(b\operatorname{rev}(a))$ and
$\Psi(b\operatorname{rev}(a))=v\operatorname{rev}(w_{S})$ where $b,v\in{\cal
A}^{n-m}$, and the claim follows from Lemma 5. ∎

###  4.3 Other lemmas

Let $\Lambda=\Lambda_{n}$ be a Young diagram growing in such a way that there
exist normalized row lengths (3). Consider its Frobenius notation:
$\Lambda=(\alpha_{1},\ldots,\alpha_{k}|\beta_{1},\ldots,\beta_{k})$. Let
$a\in{\cal A}^{m}$, and let $m_{i}$ be the multiplicity of $i$ in $a$ (thus,
$m_{1}+\ldots+m_{k}=m$). Denote by $\Lambda^{a}$ the diagram obtained from
$\Lambda$ by deleting $m_{i}$ cells from the $i$th row (if it exists).

Hereafter, by $C_{a}:=\\{y\in X:[y]_{m}=a\\}$ we denote the cylinder set of
sequences that begin with $a$. Recall that
$m_{\hat{p}}(C_{a})=\prod\hat{p}_{i}^{m_{i}}$ is the Bernoulli measure of this
set.

######  Lemma 6.

As $n\to\infty$,

$\frac{\dim\Lambda^{a}}{\dim\Lambda}\sim
m_{\hat{p}}(C_{a})\cdot\prod_{i<j}\frac{\alpha_{i}-\alpha_{j}-(m_{i}-m_{j})}{\alpha_{i}-\alpha_{j}}.$
(5)

In particular, if $\lambda_{i}-\lambda_{j}\to\infty$ for all $i<j$, then

$\lim_{n\to\infty}\frac{\dim\Lambda^{a}}{\dim\Lambda}=m_{\hat{p}}(C_{a}).$

###### Proof.

We use the formula for the dimension of a diagram in terms of its Frobenius
notation (see [3, Proposition 1.3]):

$\dim\Lambda=\frac{n!}{\prod_{i=1}^{k}\alpha_{i}!\beta_{i}!}\frac{\prod_{1\leq
i<j\leq
k}(\alpha_{i}-\alpha_{j})(\beta_{i}-\beta_{j})}{\prod_{i,j=1}^{k}(\alpha_{i}+\beta_{j}+1)}.$

Note that (for sufficiently large $n$) for
$\Lambda^{a}=(\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{k}|\beta^{\prime}_{1},\ldots,\beta^{\prime}_{k})$
we have $\alpha^{\prime}_{i}=\alpha-m_{i}$, $\beta^{\prime}_{i}=\beta_{i}$.
Thus,

$\frac{\dim\Lambda^{a}}{\dim\Lambda}=\frac{(n-m)!}{n!}\prod_{i}\frac{\alpha_{i}!}{(\alpha_{i}-m_{i})!}\prod_{i<j}\frac{\alpha_{i}-\alpha_{j}-(m_{i}-m_{j})}{\alpha_{i}-\alpha_{j}}\prod_{i,j}\frac{\alpha_{i}+\beta_{j}+1}{\alpha_{i}-m_{i}+\beta_{j}+1}.$

Since $\beta_{j}\leq k-1$ and $\frac{\alpha_{i}}{n}\to\hat{p}_{i}$ as
$n\to\infty$, the lemma follows. ∎

Denote by $\Delta(x)=\Delta(x_{1},\ldots,x_{n})$ the Vandermonde determinant
in variables $x=(x_{1},\ldots,x_{n})$. Given an arbitrary Young diagram $\mu$
with at most $n$ rows, set

$\Delta_{\mu}(x):=\sum_{\zeta}\Delta(x_{1}-\zeta_{1},\ldots,x_{n}-\zeta_{n}),$

where the sum is over all different permutations $\zeta$ of the sequence
$\mu=(\mu_{1},\mu_{2},\ldots,\mu_{n})$ (padded with $0$s if necessary).

######  Lemma 7.

$\Delta_{\mu}(x)=n_{\mu}\Delta(x)$, where $n_{\mu}$ is the number of different
permutations of $\mu$.

###### Proof.

It suffices to prove that

$\sum_{\sigma\in{\mathfrak{S}}_{n}}\Delta(x_{1}-\mu_{\sigma(1)},\ldots,x_{n}-\mu_{\sigma(n)})=n!\Delta(x),$

where ${\mathfrak{S}}_{n}$ is the symmetric group, since the sum in the left-
hand side differs from $\Delta_{\mu}(x)$ exactly by the factor
$\frac{n!}{n_{\mu}}$. If the length $\ell$ of $\mu$ is equal to $1$, this is a
well-known identity for Vandermonde determinants (see, e.g., [5, Proposition
7.192]):

$\sum_{i=1}^{n}\Delta(x_{1},\ldots,x_{i-1},x_{i}-t,x_{i+1},\ldots,x_{n})=n\Delta(x).$

Applying it successively several times, one can easily prove the required
identity by induction. ∎

###  4.4 Proof of Thoma’s theorem

By Lemma 2, it suffices to prove that every ergodic central measure on the
graph $\operatorname{SW}_{k}$ projects to a Thoma measure.

We apply the ergodic method. Let $x\in X$, and assume that there exist
densities (2), and hence, by Proposition 2, normalized row lengths (3). To
simplify notation, we will assume that $p_{1}\geq p_{2}\geq\ldots\geq p_{k}$;
in the general case, the argument remains the same, with $p_{i}$ everywhere
replaced by $\hat{p}_{i}$. Denote by ${\mathfrak{C}}_{n}(x)$ the class of the
Young tail partition $\bar{\eta}_{n}$ containing $[x]_{n}$, and let
$\mu_{x}^{(n)}$ be the uniform measure on ${\mathfrak{C}}_{n}(x)$. Set
$T_{n}=P([x]_{n})$ and $\Lambda_{n}=\operatorname{sh}(T_{n})$.

Fix $m\in{\mathbb{N}}$ and $a\in{\cal A}^{m}$. We have

$\mu_{x}(C_{a})=\lim_{n\to\infty}\mu_{x}^{(n)}(C_{a})=\lim_{n\to\infty}\frac{c_{a}(x,n)}{\\#{\mathfrak{C}}_{n}(x)}=\lim_{n\to\infty}\frac{c_{a}(x,n)}{\dim\Lambda_{n}},$

where $c_{a}(x,n)=\\#\\{y\in{\mathfrak{C}}_{n}(x):[y]_{m}=a\\}$.

Let $C_{\lambda}$ be the cylinder set of paths in the Young graph passing
through a vertex $\lambda$. It suffices to prove that for every Young diagram
$\lambda$ with $m$ cells,

$\sum_{a\colon\operatorname{sh}(a)=\lambda}\mu_{x}(C_{a})={\cal
M}_{(p,0,0)}(C_{\lambda})=\dim\lambda\cdot s_{\lambda}(p),$ (6)

where $s_{\lambda}$ is a Schur function.

Given a word $w\in{\cal A}^{m}$, let $\Lambda_{n}^{w}$ be the diagram obtained
from $\Lambda_{n}$ by successively deleting a cell from the $w_{1}$th,
$w_{2}$th, … row (if possible). In the notation of Lemma 6, consider an
arbitrary subsequence of indices $n$ along which all differences
$\alpha_{i}-\alpha_{j}$ have a (finite or infinite) limit, and set

$\phi(w):=\lim_{n\to\infty}\frac{\dim\Lambda^{w}_{n}}{\dim\Lambda_{n}}.$

If cells cannot be deleted in this way, set $\phi(w)=0$. Also set
$\Phi_{T_{n}}(w):=\Phi_{T_{n}}(S_{w})$, where $S_{w}\in{\cal
S}_{m}(\Lambda_{n})$ is the tableau corresponding to this sequence of deleted
cells. It follows from Lemma 4 that

$\mu_{x}(C_{a})=\sum_{w\colon\Phi_{T_{n}}(w)=a}\phi(w),$

and Corollary 2 of Lemma 5 implies that

$\sum_{a\colon\operatorname{sh}(a)=\lambda}\mu_{x}(C_{a})=\sum_{w\colon\operatorname{sh}(w)=\lambda}\phi(w).$

First, assume that

$\lambda_{i}-\lambda_{j}\to\infty\quad\text{for any}\quad i<j$ (7)

(note that this condition is automatically satisfied for given $i,j$ if
${p_{i}\neq p_{j}}$). Then $\alpha_{i}-\alpha_{j}\to\infty$ for any $i<j$, and
by Lemma 6 we have $\phi(w)=\prod_{i}p_{i}^{m_{i}(w)}$, where $m_{i}(w)$ is
the multiplicity of $i$ in $w$. It follows that

$\sum_{a\colon\operatorname{sh}(a)=\lambda}\mu_{x}(C_{a})=\sum_{w\colon\operatorname{sh}(w)=\lambda}\prod_{i}p_{i}^{m_{i}(w)}=\dim\lambda\cdot
s_{\lambda}(p)={\cal M}_{(p,0,0)}(C_{\lambda});$

here we have used the combinatorial definition of Schur functions

$s_{\lambda}(p)=\sum_{S\colon\operatorname{sh}(S)=\lambda}\prod_{i}p_{i}^{m_{i}(S)},$
(8)

where $m_{i}(S)$ is the multiplicity of $i$ in a semistandard tableau $S$, and
the fact that for every tableau $S$ of shape $\lambda$ there are exactly
$\dim\lambda$ words $w$ such that $P(w)=S$. Thus, in the case under
consideration equality (6) is proved.

Now assume that condition (7) is not satisfied. First, consider the case where
it fails for all pairs $i<j$, and hence ${p_{1}=\ldots=p_{k}=\frac{1}{k}}$. By
Lemma 6,

$\phi(w)=\frac{1}{k^{m}}\cdot\prod_{i<j}\frac{\alpha_{i}-\alpha_{j}-(m_{i}-m_{j})}{\alpha_{i}-\alpha_{j}}=\frac{1}{k^{m}\Delta(\alpha)}\cdot\Delta(\alpha_{1}-m_{1},\ldots,\alpha_{k}-m_{k}).$

Thus, the sum we are interested in is equal to

$\frac{1}{k^{m}\Delta(\alpha)}\sum_{w\colon\operatorname{sh}(w)=\lambda}\Delta(\alpha_{1}-m_{1},\ldots,\alpha_{k}-m_{k}).$

Since the sum (8) defines a symmetric function, it follows that the last sum
is a linear combination of determinants of the form $\Delta_{\mu}(\alpha)$,
and hence, by Lemma 7,

$\sum_{w\colon\operatorname{sh}(w)=\lambda}\phi(w)=\frac{\dim\lambda\cdot
d_{\lambda}}{k^{m}},$

where $d_{\lambda}$ is the number of different semistandard tableaux of shape
$\lambda$. The right-hand side of the last equality is equal to
$\dim\lambda\cdot s_{\lambda}(\frac{1}{k},\ldots,\frac{1}{k})$, so equality
(6) is proved in this case too.

Finally, consider the case where there are several families of equal
densities. Namely, let $q_{1},\ldots,q_{s}$ be all different values of
densities and $J_{j}:=\\{i\colon p_{i}=q_{j}\\}$, $j=1,\ldots,s$. Set
$n_{j}=\sum_{i\in J_{j}}m_{i}$. By Lemma 6,

$\phi(w)=\prod_{j=1}^{s}\frac{q_{j}^{n_{j}}}{\Delta_{j}(\alpha)}\prod_{j=1}^{s}\Delta_{j}(\alpha-m),$

where $\Delta_{j}(\alpha)$ and $\Delta_{j}(\alpha-m)$ are the Vandermonde
determinants in the variables $\alpha_{i}$, $i\in J_{j}$, and
$\alpha_{i}-m_{i}$, $i\in J_{j}$, respectively. We are interested in the sum
$\sum\limits_{w\colon\operatorname{sh}(w)=\lambda}\phi(w)$. Fix the block
multiplicities $n_{1},\ldots,n_{s}$ and consider the sum over all words $w$
with these multiplicities. Similarly to the argument above, we conclude that,
first, this sum splits into the product of sums over separate blocks and,
second, the sum of the determinants $\Delta_{j}(\alpha-m)$ in each block is
equal to $\Delta_{j}(\alpha)$ with an appropriate factor by Lemma 7. The
result is again (6).

Thus, if $x\in X$ is a sequence such that the densities (2) exist, the weak
limit $\mu_{x}$ of the measures $\mu_{x}^{(n)}$ projects to the Thoma measure
${\cal M}_{(p,0,0)}$ under the projection of the Schur–Weyl graph onto the
Young graph. It follows (by considering convergent subsequences) that if the
densities (2) do not exist, then the measures $\mu_{x}^{(n)}$ have no limit.

The theorem is proved.

Remarks. We say that a central measure $\mu$ on the Schur–Weyl graph is
nondegenerate if $\mu(C_{a})\neq 0$ for every finite word $a$. One can easily
see from the proof that a nondegenerate ergodic central measure with given
densities on the Schur–Weyl graph is unique and coincides with the image of
the corresponding Bernoulli measure under the isomorphism from Lemma 1. Apart
from this measure, there exists a family of degenerate central measures,
supported on tableaux with restrictions on the number of “free” elements
(defined like in the proof of Lemma 3). In particular, the measure constructed
in Lemma 2 is the “most degenerate” measure, supported on tableaux with no
free elements.

##  5 The general finite case

Let ${\cal A}_{k}:=\\{1,\ldots,k\\}$ and denote by ${\cal
A}^{*}_{\ell}:=\\{i^{*}\colon i\in{\cal A}_{\ell}\\}$ the alphabet opposite to
${\cal A}_{\ell}$, in which the order is defined as ${1^{*}>\ldots>\ell^{*}}$.
Consider the alphabet ${\widetilde{A}}={\cal A}_{k}\cup{\cal A}^{*}_{\ell}$
with the order determined by the orders on ${\cal A}_{k}$ and ${\cal
A}^{*}_{\ell}$ and the convention that $a<b$ if $a\in{\cal A}_{k}$, $b\in{\cal
A}^{*}_{\ell}$. A Bernoulli measure $m_{\alpha,\beta}$ on ${\widetilde{A}}$ is
determined by vectors $\alpha=(\alpha_{1},\ldots,\alpha_{k})$ and
$\beta=(\beta_{1},\ldots,\beta_{\ell})$, where
$\alpha_{i}=\operatorname{Prob}(i)$, $\beta_{j}=\operatorname{Prob}(j^{*})$.

Following [8], consider the algorithm $\widetilde{\operatorname{RSK}}$ that
acts on sequences ${x\in{\widetilde{A}}^{\infty}}$ as follows: it applies the
ordinary row insertion to symbols from ${\cal A}$ (“row symbols”) and the dual
row insertion to symbols from ${\cal A}^{*}$ (“column symbols”). Set
$\widetilde{\operatorname{RSK}}(x)=(\widetilde{P}(x),\widetilde{Q}(x))$. Here
$\widetilde{Q}(x)$ is still a standard tableau, while $\widetilde{P}(x)$ is a
$(k,\ell)$-semistandard tableau (see [2]).

As shown in [8], in this case the youngization (see Definition 1) is again a
homomorphism (and even, as proved in [12], an isomorphism) of measure spaces
$\pi:({\widetilde{A}}^{\infty},m_{\alpha,\beta}^{\infty})\to({\cal
T}({\mathbb{Y}}),{\cal M}_{(\alpha,\beta,0)})$, where ${\cal
M}_{(\alpha,\beta,0)}$ is the central measure on ${\cal T}({\mathbb{Y}})$ with
Thoma parameters $(\alpha,\beta,0)$.

In this case, we introduce the Schur–Weyl graph $\operatorname{SW}$ similarly
to Definition 3. It is easy to see that Lemma 2 remains valid (to obtain the
“maximal” tableau with given diagram, we fill its first $k$ rows with elements
of ${\cal A}_{k}$ in the same way as before; the remaining part is of shape
$\nu^{t}$ where $\nu$ is a diagram with at most $\ell$ rows, and we fill $\nu$
with elements of ${\cal A}^{*}_{\ell}$ in a similar way and transpose). Hence,
to prove Thoma’s theorem in the general finite case, we can use the same
approach as above: find the central measures on the Schur–Weyl graph by the
ergodic method and prove that they project to Thoma measures. To reduce the
proof of the general case to that considered above, we use the following
lemma.

######  Lemma 8.

Given a word $w\in{\widetilde{A}}^{\infty}$, denote by $w^{\dagger}$ the word
obtained from it by replacing each symbol $i$ with $i^{*}$ and each symbol
$j^{*}$ with $j$. Then $\widetilde{Q}(w^{\dagger})=\widetilde{Q}(w)^{t}$.

###### Proof.

It suffices to prove that
$\operatorname{sh}(\widetilde{Q}(w^{\dagger}))=\operatorname{sh}(\widetilde{Q}(w))^{t}$
for every word ${w\in{\widetilde{A}}^{\infty}}$. Given
$a,b\in{\widetilde{A}}$, we write $a\nearrow b$ if $a<b$ or $a=b\in{\cal
A}_{k}$, and $a\searrow b$ if $a>b$ or $a=b\in{\cal A}^{*}_{\ell}$. A word
$u_{1}\ldots u_{s}\in{\widetilde{A}}$ is said to be increasing (decreasing) if
$u_{i}\nearrow u_{i+1}$ (respectively, $u_{i}\searrow u_{i+1}$) for all
$i=1,\ldots,s-1$. Clearly, $u$ is increasing (decreasing) if and only if
$u^{\dagger}$ is decreasing (increasing). Now the lemma follows from [8,
Proposition 1], according to which the length of the first row (column) of the
diagram $\operatorname{sh}(\widetilde{Q}(w))$ is the length of the largest
increasing (decreasing) subsequence in $w$, the sum of the lengths of the two
first rows (columns) is the maximum cardinality of the union of two increasing
(decreasing) subsequences in $w$, etc. ∎

Given $x\in{\widetilde{A}}^{\infty}$, denote by
$\lambda_{1}^{\prime(n)},\ldots,\lambda_{\ell}^{\prime(n)}$ the column lengths
of the diagram $\operatorname{sh}([x]_{n})$. In addition to the densities (2)
and the normalized row lengths (3), consider the densities

$q_{j}:=\lim_{n\to\infty}\frac{\\#\\{i\leq n:x^{*}_{i}=j^{*}\\}}{n},\quad
j^{*}=1^{*},\ldots,\ell^{*},$ (9)

and the normalized column lengths

$\lim_{n\to\infty}\frac{\lambda_{j}^{\prime(n)}}{n}=\hat{q}_{j},\quad
j^{*}=1^{*},\ldots,\ell^{*}.$ (10)

######  Proposition 3.

If there exist densities (2) and (9), then there exist normalized row and
column lengths (3) and (10), where $\hat{p}_{1},\ldots,\hat{p}_{k}$
(respectively, $\hat{q}_{1},\ldots,\hat{q}_{\ell}$) are the densities
$p_{1},\ldots,p_{k}$ (respectively, $q_{1},\ldots,q_{\ell}$) arranged in
nonincreasing order.

###### Proof.

Assume that the densities (2) and (9) exist. Since elements from ${\cal
A}^{*}_{\ell}$ do not affect the dynamics of the subtableau consisting of
elements from ${\cal A}_{k}$, it follows from Proposition 2 that
$\varliminf\limits_{n\to\infty}\frac{\lambda_{j}^{(n)}}{n}\geq\hat{p}_{j}$.
Applying Lemma 8, we see that
$\varliminf\limits_{n\to\infty}\frac{\lambda_{j}^{\prime(n)}}{n}\geq\hat{q}_{j}$.
The remaining part of the proof is obvious. ∎

Then the proof follows the same scheme as for Theorem 2 with natural changes.
Recall (see, e.g., [20]) that the cylinder distributions of the Thoma measure
${\cal M}_{(\alpha,\beta,0)}$ are given by the formula

${\cal
M}_{(\alpha,\beta,0)}(C_{\lambda})=\dim\lambda\cdot\tilde{s}_{\lambda}(\alpha,\beta),$

where $\tilde{s}_{\lambda}(\alpha,\beta)$ are generalized (“hook”) Schur
functions. For them, an analog of formula (8) holds in which the sum is over
$(k,\ell)$-semistandard tableaux (see [2]). It is not difficult to show that
identity (4) takes the form

$\widetilde{\operatorname{RSK}}(w)=(P,Q)\iff\widetilde{\operatorname{RSK}}^{*}(\operatorname{rev}(w))=(P^{t},\operatorname{evac}(Q^{t})),$

where the $\widetilde{\operatorname{RSK}}^{*}$ algorithm applies the ordinary
row insertion to column symbols and the dual one to row symbols. When
calculating $\Phi_{T}(S)$, at each step we apply $\varepsilon^{*}$ and
$\tau^{*}$ for row symbols and $\varepsilon$ and $\tau$ for column symbols;
then Lemma 4 remains without change. When calculating $\Psi(u)$, we set
$u^{\prime}_{i}$ to be the number of the column (respectively, row) to which a
cell is added at the $i$th step of the algorithm
$\widetilde{\operatorname{RSK}}^{*}(u)$ if $u_{i}\in{\cal A}_{k}$
(respectively, $u_{i}\in{\cal A}^{*}_{\ell}$); then Lemma 5 (with $Q$ replaced
by $\widetilde{Q}$) and Corollary 2 remain valid. In the asymptotics from
Lemma 6, similar factors corresponding to columns appear, and the remaining
part of the proof goes without significant changes.

Thus, we obtain Thoma’s theorem for the general finite case.

######  Theorem 3.

All ergodic central measures on the Young graph supported on tableaux with at
most $k$ rows and at most $\ell$ columns are exhausted by the Thoma measures
$\mu_{(\alpha,\beta,0)}$.

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