The Martin Boundary of the Young-Fibonacci Lattice.
In this paper we find the Martin boundary for the Young-Fibonacci lattice YF. Along with the lattice of Young diagrams, this is the most interesting example of a differential partially ordered set. The Martin boundary construction provides an explicit Poisson-type integral representation of non-negative harmonic functions on YF. The latter are in a canonical correspondence with a set of traces on the locally semisimple Okada algebra. The set is known to contain all the indecomposable traces. Presumably, all of the traces in the set are indecomposable, though we have no proof of this conjecture. Using an explicit product formula for Okada characters, we derive precise regularity conditions under which a sequence of characters of finite-dimensional Okada algebras converges.
Introduction
The Young-Fibonacci lattice YF is a fundamental example of a differential partially
ordered set which was introduced by R. Stanley [St1] and S. Fomin [F1]. In many
ways, it is similar to another major example of a differential poset, the Young lattice
Y. Addressing a question posed by Stanley, S. Okada has introduced [Ok] two algebras
associated to YF. The first algebra F is a locally semisimple algebra defined by generators
and relations, which bears the same relation to the lattice YF as does the group
algebra C S1 of the infinite symmetric group to Young's lattice. The second algebra
R is an algebra of non-commutative polynomials, which bears the same relation to the
lattice YF as does the ring of symmetric functions to Young's lattice.
The purpose of the present paper is to study some combinatorics, both finite and
asymptotic, of the lattice YF. Our object of study is the compact convex set of harmonic
functions on YF (or equivalently the set of positive normalized traces on F or
certain positive linear functionals on R.) We address the study of harmonic functions
by determining the Martin boundary of the lattice YF. The Martin boundary is the
set consisting of those harmonic functions which can be obtained by finite
rank approximation. There are two basic facts related to the Martin boundary con-
struction: 1) every harmonic function is represented by the integral of a probability
measure on the Martin boundary, and 2) the set of extreme harmonic functions is a
subset of the Martin boundary (see, e.g., [D]).
This paper gives a parametrization of the Martin boundary for YF and a description
of its topology.
The Young-Fibonacci lattice is described in Section 2, and preliminaries on harmonic
functions are explained in Section 3. A first rough description of our main results is
given at the end of Section 3. precise description of the parametrization of harmonic
functions is found in Section 7, and the proof, finally, is contained in Section 8.) Section
contains some general results on harmonic functions on differential posets.
The main tool in our study is the Okada ring R and two bases of this ring, introduced
by Okada, which are in some respect analogous to the Schur function basis and the
power sum function basis in the ring of symmetric functions (Section 5). We describe
the Okada analogs of the Schur function basis by non-commutative determinants of
tridiagonal matrices with monomial entries. We obtain a simple and explicit formula
for the transition matrix (character matrix) connecting the s-basis and the p-basis,
and also for the value of (the linear extension of) harmonic functions evaluated on the
p-basis. This is done in Sections 6 and 7.
The explicit formula allows us to study the regularity question for the lattice YF, that
is the question of convergence of extreme traces of finite dimensional Okada algebras Fn
to traces of the inductive limit algebra . The regularity question is studied
in Section 8.
The analogous questions for Young's lattice Y(which is also a differential poset) were
answered some time ago. The parametrization of the Martin boundary of Y has been
studied in [Th], [VK]. A different approach was recently given in [Oku].
A remaining open problem for the Young-Fibonacci lattice is to characterize the set
of extreme harmonic functions within the Martin boundary. For Young's lattice, the
set of extreme harmonic functions coincides with the entire Martin boundary.

Acknowledgement

. The second author thanks the Department of Mathematics, University
of Iowa, for a teaching position in the Spring term of 1993. Most part of the
present work was completed during this visit.
4 FREDERICK M. GOODMAN AND SERGEI V. KEROV
x2. The Young-Fibonacci lattice
In this Section we recall the definition of Young-Fibonacci modular lattice (see Figure
1) and some basic facts related to its combinatorics. See Section A.1 in the Appendix
for the background definitions and notations related to graded graphs and differential
posets. We refer to [F1-2], [St1-3] for a more detailed exposition.
A simple recurrent construction.
The simplest way to define the graded graph
n=0 YFn is provided by the
following recurrent procedure.
Let the first two levels YF 0 and YF 1 have just one vertex each, joined by an edge.
Assuming that the part of the graph YF, up to the nth level YFn , is already constructed,
we define the set of vertices of the next level YFn+1 , along with the set of adjacent edges,
by first reflecting the edges in between the two previous levels, and then by attaching
just one new edge leading from each of the vertices on the level YFn to a corresponding
new vertex at level n + 1.
@
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@
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@
@
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A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
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Figure

1. The Young-Fibonacci lattice.
In particular, we get two vertices in the set YF 2 , and two new edges: one is obtained
by reflecting the only existing edge, and the other by attaching a new one. More
generally, there is a natural notation for new vertices which helps to keep track of the
inductive procedure. Let us denote the vertices of YF 0 and YF 1 by an empty word ?
and 1 correspondingly. Then the endpoint of the reflected edge will be denoted by 2,
and the end vertex of the new edge by 11. In a similar way, all the vertices can be
labeled by words in the letters 1 and 2. If the left (closer to the root ?) end of an edge
is labeled by a word v, then the endvertex of the reflected edge is labeled by the word
2v. Each vertex w of the nth level is joined to a vertex 1w at the next level by a new
edge (which is not a reflection of any previous edge).
Clearly, the number of vertices at the nth level YFn is the nth Fibonacci number fn .
Basic definitions.
We now give somewhat more formal description of the Young-Fibonacci lattice and
its Hasse diagram.
Definition. A finite word in the two-letter alphabet f1; 2g will be referred to as a
Fibonacci word. We denote the sum of digits of a Fibonacci word w by jwj, and we call
it the rank of w. The set of words of a given rank n will be denoted by YFn , and the set
of all Fibonacci words by YF. The head of a Fibonacci word is defined as the longest
contiguous subword of 2's at its left end. The position of a 2 in a Fibonacci word is one
more than the rank of the subword to the right of the 2; that is if then the
position of the indicated 2 is jvj + 1.
Next we define a partial order on the set YF which is known to make YF a modular
lattice. The order will be described by giving the covering relations on YF in two
equivalent forms.
Given a Fibonacci word v, we first define the set v ae YF of its successors. By
definition, this is exactly the set of words w 2 YF which can be obtained from v by one
of the following three operations:
(i) put an extra 1 at the left end of the word v;
(ii) replace the first 1 in the word v (reading left to right) by 2;
anywhere in between 2's in the head of the word v, or immediately after
the last 2 in the head.
Example. Take 222121112 for the word v of rank 14. Then the group of 3 leftmost 2's
forms its head, and v has 5 successors, namely
The changing letter is shown in boldface. Note that the ranks of all successors of a
Fibonacci word v are one bigger than that of v.
The set v of predecessors of a non-empty Fibonacci word v can be described in a
similar way. The operations to be applied to v in order to obtain one of its predecessors
are as follows:
(i) the leftmost letter 1 in the word v can be removed;
(ii) any one of 2's in the head of v can be replaced by 1.
6 FREDERICK M. GOODMAN AND SERGEI V. KEROV
Example. The word predecessors, namely
We write u % v to show that v is a successor of u (and u is a predecessor of v).
This is a covering relation which determines a partial order on the set YF of Fibonacci
words. As a matter of fact, it is a modular lattice, see [St1]. The initial part of the
Hasse diagram of the poset YF is represented in Figure 1.
The Young-Fibonacci lattice as a differential poset.
Assuming that the head length of v is k, the word v has
predecessors, if v contains at least one letter 1. If is made of 2's only, it
predecessors. Note that the number of successors is always
one bigger than that of predecessors. Another important property of the lattice YF is
that, for any two different Fibonacci words v 1 , v 2 of the same rank, the number of their
common successors equals that of common predecessors (both numbers can only be 0 or
1). These are exactly the two characteristic properties (D1), (D2) of differential posets,
see Section A.1. In what follows we shall frequently use the basic facts on differential
posets, surveyed for the readers convenience in the Appendix. Much more information
on differential posets and their generalizations can be found in [F1], [St1].
The Okada algebra.
Okada [Ok] introduced a (complex locally semisimple) algebra F , defined by generators
and relations, which admits the Young-Fibonacci lattice YF as its branching
diagram. The Okada algebra has generators (e i ) i-1 satisfying the relations:
2.
The algebra Fn generated by the first n\Gamma1 generators e these identities
is semisimple of dimension n!, and has simple modules M v labelled by elements
For one has u % v if, and only if, the simple Fn-module M v ,
restricted to the algebra Fn\Gamma1 contains the simple Fn\Gamma1 -module Mu . As a matter of
fact, the restrictions of simple Fn-modules to Fn\Gamma1 are multiplicity free.
x3. Harmonic functions on graphs and traces of AF-algebras
In this Section, we recall the notion of harmonic functions on a graded graph and the
classical Martin boundary construction for graded graphs and branching diagrams. We
discuss the connection between harmonic functions on branching diagrams and traces
on the corresponding AF -algebra. Finally, we give a preliminary statement on our main
results on the Martin boundary of the Young-Fibonacci lattice.
We refer the reader to Appendix A.1 for basic definitions on graded graphs and
branching diagrams and to [E], [KV] for more details on the combinatorial theory of
AF -algebras.
The Martin boundary of a graded graph.
A function R defined on the set of vertices of a graded graph \Gamma is called
harmonic, if the following variant of the "mean value theorem" holds for all vertices
w:u%w
We are interested in the problem of determining the space H of all non-negative harmonic
functions normalized at the vertex ? by the condition
compact convex set with the topology of pointwise convergence, it is interesting to ask
about its set of extreme points.
A general approach to the problem of determining the set of extreme points is based
on the Martin boundary construction (see, for instance, [D]). One starts with the dimension
function d(v; w) defined as the number of all oriented paths from v to w. We
put
From the point of view of potential theory, d(v; w) is the Green function with respect
to "Laplace operator"
w:u%w
This means that if /w fixed vertex w, then \Gamma(\Delta/ w
\Gamma. The ratio
is usually called the Martin kernel.
Consider the space Fun(\Gamma) of all functions with the topology of pointwise
convergence, and let e
E be the closure of the subset e
ae Fun(\Gamma) of functions v 7! K(v; w),
those functions are uniformly bounded, 0 - K(v; w) - 1, the space e
(called the Martin compactification) is indeed compact. One can easily check that e \Gamma ae e
is a dense open subset of e
Its boundary
called the Martin boundary of
the graph \Gamma.
By definition, the Martin kernel (3.3) may be extended by continuity to a function
R. For each boundary point
negative, harmonic, and normalized. Moreover, harmonic functions have an integral
representation similar to the classical Poisson integral representation for non-negative
harmonic functions in the disk:
8 FREDERICK M. GOODMAN AND SERGEI V. KEROV
Theorem (cf. [D]). Every normalized non-negative harmonic function ' 2 H admits
an integral representation
Z
where M is a probability measure. Conversely, for every probability measure M on E,
the integral (3.4) provides a non-negative harmonic function ' 2 H.
All indecomposable (i.e., extreme) functions in H can be represented in the form
and we denote by Emin the
corresponding subset of the boundary E. It is known that Emin is a non-empty G ffi
subset of E. One can always choose the measure M in the integral representation
(3.4) to be supported by Emin . Under this assumption, the measure M representing a
function
Given a concrete example of a graded graph, one looks for an appropriate "geometric"
description of the abstract Martin boundary. The purpose of the present paper is to
give an explicit description for the Martin boundary of the Young - Fibonacci graph
YF.
The traces on locally semisimple algebras.
We next discuss the relation between harmonic functions on a graded graph and
traces on locally semisimple algebras. A locally semisimple complex algebra A (or AF-
algebra) is the union of an increasing sequence of finite dimensional semisimple complex
algebras, An . The branching diagram or Bratteli diagram \Gamma(A) of a locally
semisimple algebra A (more precisely, of the approximating sequence fAng) is a graded
graph whose vertices of rank n correspond to the simple An-modules. Let M v denote
the simple An-module corresponding to a vertex . Then a vertex v of rank n
and a vertex w of rank are joined by -(v; w) edges if the simple An+1 module
Mw , regarded as an An module, contains M v with multiplicity -(v; w). We will assume
here that all multiplicities -(v; w) are 0 or 1, as this is the case in the example of
the Young-Fibonacci lattice with which we are chiefly concerned. Conversely, given a
branching diagram \Gamma - that is, a graded graph with unique minimal vertex at rank 0
and no maximal vertices - there is a locally semisimple algebra A such that
A trace on a locally semisimple algebra A is a complex linear functional / satisfying
To each trace / on A, there corresponds a positive normalized harmonic function ~
/ on
whenever v has rank n and e is a minimal idempotent in An such that eM v 6= (0)
and fvg. The trace property of / implies that ~
/ is a well
defined non-negative function on \Gamma, and harmonicity of ~
/ follows from the definition of
the branching diagram \Gamma(A). Conversely, a positive normalized harmonic function ~
/ on
defines a trace on A; in fact, a trace on each An is determined by its value
on minimal idempotents, so the assignment
whenever e is a minimal idempotent in An such that eM v 6= (0), defines a trace on
An . The harmonicity of ~
/ implies that the / (n) are coherent, i.e., the restriction of
/ (n+1) from An+1 to the subalgebra An coincides with / (n) . As a result, the traces / (n)
determine a trace of the limiting algebra
The set of traces on A is a compact convex set, with the topology of pointwise
convergence. The map ~
is an affine homeomorphism between the space of positive
normalized harmonic functions on and the space of traces on A. From the
point of view of traces, the Martin boundary of \Gamma consists of traces / which can be
obtained as limits of a sequence /n , where /n is an extreme trace on An . All extreme
traces on A are in the Martin boundary, so determination of the Martin boundary is a
step towards determining the set of extreme traces on A.
The locally semisimple algebra corresponding to the Young-Fibonacci lattice YF is
the Okada algebra F introduced in Section 2.
The main result.
We can now give a description of the Martin boundary of the Young-Fibonacci lattice
(and consequently of a Poisson-type integral representation for non-negative harmonic
functions on YF).
Let w be an infinite word in the alphabet f1; 2g (infinite Fibonacci word),
and let d the positions of 2's in w. The word w is said to be summable
if, and only if, the series
equivalently, the product
Y
converges.
As for any differential poset, the lattice YF has a distinguished harmonic function
'P , called the Plancherel harmonic function; 'P is an element of the Martin boundary.
The complement of f'P g in the Martin boundary of YF can be parametrized with two
parameters (fi; w); here fi is a real number, 0 ! fi - 1, and w is a summable infinite
word in the alphabet f1; 2g.
We denote
by\Omega the parameter space for the Martin boundary:
Definition. Let the
space\Omega be the union of a point P and the set
summable infinite word in the alphabet f1; 2gg;
with the following topology: A sequence (fi (n) ; w (n) ) converges to P iff
A sequence (fi (n) ; w (n) ) converges to (fi; w) if, and only if,
We will describe in Section 7 the mapping ! 7! '!
from\Omega to the set of normalized
positive harmonic functions on YF.
We are in a position now to state the main result of the paper.
Theorem 3.2. The map ! 7! '! is a homeomorphism of the
space\Omega onto the Martin
boundary of the Young-Fibonacci lattice. Consequently, for each probability measure M
on\Omega , the integral
Z
provides a normalized, non-negative harmonic function on the Young-Fibonacci lattice
YF. Conversely, every such function admits an integral representation with respect to a
measure M
on\Omega (which may not be unique).
In general, for all differential posets, we show that there is a flow
on [0; 1] \Theta H with the properties
ts (') and C 0
For the Young-Fibonacci lattice, one has C t (' fi;w In
particular, the flow on H preserves the Martin boundary. It is not clear whether this is
a general phenomenon for differential posets.
We have not yet been able to characterize the extreme points within the Martin
boundary of YF. In a number of similar examples, for instance the Young lattice, all
elements of the Martin boundary are extreme points.
x4. Harmonic functions on differential posets
The Young-Fibonacci lattice is an example of a differential poset. In this section,
we introduce some general constructions for harmonic functions on a differential poset.
Later on in Section 7 we use the construction to obtain the Martin kernel of the graph
YF.
Type I harmonic functions.
In this subsection we don't need any special assumptions on the branching diagram
\Gamma. Consider an infinite path
in \Gamma. For each vertex the sequence fd(u; vn )g 1
n=1 is weakly increasing, and we
shall use the notation
Note that d(u; if the sequences t; s coincide eventually.
Lemma 4.1. The following conditions are equivalent for a path t in \Gamma:
(i) All but finitely many vertices vn in the path t have the single immediate predecessor
.
(iv) There are only finitely many paths which eventually coincide with t.
Proof. It is clear that d(?; is the only predecessor of vn . Since
(i). The number of paths
equivalent to t is exactly d(?; t).
In case is the Young lattice, there are only two paths (i.e. Young tableaux) with
these conditions: In case of Young-
Fibonacci lattice there are countably many paths satisfying the conditions of Lemma
4.1. The vertices of such a path eventually take the form
Fibonacci word v of rank m. Hence, the equivalence class of eventually coinciding paths
in YF with the properties of Lemma 4.1 can be labelled by infinite words in the alphabet
f2; 1g with only finite number of 2's. We denote the set of such words as 1 1 YF.
Proposition 4.2. Assume that a path t in \Gamma satisfies the conditions of Lemma 4.1.
Then
is a positive normalized harmonic function on \Gamma.
Proof. Since
w:v%w d(w; t), the function ' t is harmonic. Also, ' t (v) - 0 for
all
We say that these harmonic functions are of type I, since the corresponding AF -
algebra traces are traces of finite - dimensional irreducible representations (type I factor -
representations). It is clear that all the harmonic functions of type I are indecomposable.
Plancherel harmonic function.
Let us assume now that the poset \Gamma is differential in the sense of [St1] or, equivalently,
is a self-dual graph in terms of [F1]. The properties of differential posets which we need
are surveyed in the Appendix.
Proposition 4.3. The function
is a positive normalized harmonic function on the differential poset \Gamma.
Proof. This follows directly from (A.2.1) in the Appendix.
Note that if is the Young lattice, the function 'P corresponds to the Plancherel
measure of the infinite symmetric group (cf. [KV]).
Contraction of harmonic functions on a differential poset.
Assume that \Gamma is a differential poset. We shall show that for any harmonic function
' there is a family of affine transformations, with one real parameter - , connecting the
Plancherel function 'P to '.
Proposition 4.4. For 0 - 1 and a harmonic function ', define a function C - (')
on the set of vertices of the differential poset \Gamma by the formula
juj=k
Then C - (') is a positive normalized harmonic function, and the map ' 7! C t (') is
affine.
Proof. We introduce the notation
juj=k
First we observe the identity
w:v%w
which is obtained from a straightforward computation using (A.2.3) from the Appendix,
and the harmonic property (3.1) of the function '. From this we derive that
w:v%w
w:v%w
This shows that C - (') is harmonic. It is easy to see that C - (') is normalized and
positive, and that the map ' 7! C t (') is affine.
Remarks. (a) The semigroup property holds: C t (C s st ('); (b) C 0
'P , for all ', and C t ('P These statements
can be verified by straightforward computations.
Example. Let ' denote the indecomposable harmonic function on the Young lattice
with the Thoma parameters (ff; fi; fl), see [KV] for definitions. Then the function C - (')
is also indecomposable, with the Thoma parameters (- ff; -
Central measures and contractions.
Recall (see [KV]) that for any harmonic function ' on \Gamma there is a central measure
M ' on the space T of paths of \Gamma, determined by its level distributions
In particular,
There is a simple probabilistic description of the central measure corresponding to
a harmonic function on a differential poset obtained by the contraction of Proposition
4.4. Define a random vertex by the following procedure:
(a) Choose a random k, 0 - k - n with the binomial distribution
(b) Choose a random vertex
(c) Start a random walk at the vertex u, with the Plancherel transition probabilities
Let v denote the vertex at which the random walk first hits the n'th level set \Gamma n . We
denote by M (-;')
n the distribution of the random vertex v.
14 FREDERICK M. GOODMAN AND SERGEI V. KEROV
Proposition 4.5. The distribution M (-;')
n is the n'th level distribution of the central
measure corresponding to the harmonic function C - ('):
Proof. It follows from (A.2.1) that (4.10) is a probability distribution. By Lemma A.3.2,
the probability to hit \Gamma n at the vertex v, starting the Plancherel walk at
The Proposition now follows from the definition of the contraction C - (') written in the
Example. Y be the Young lattice and let be the one-
row Young tableau. Then the distribution (4.9) is trivial, and the procedure reduces
to choosing a random row diagram (k) with the distribution (4.8) and applying the
Plancherel growth process until the diagram will gain n boxes.
x5. Okada clone of the symmetric function ring
In this Section we introduce the Okada variant of the symmetric function algebra,
and its two bases analogous to the Schur function basis and the power sum basis. The
Young-Fibonacci lattice arises in a Pieri-type formula for the first basis.
The rings R and
denote the ring of all polynomials in two non-commuting
variables X;Y . We endow R with a structure of graded ring,
n=0 Rn , by declaring
the degrees of variables to be deg 2. For each word
let h v denote the monomial
Then Rn is a Q-vector space with the fn (Fibonacci number) monomials h v as a basis.
We let denote the inductive limit of linear spaces Rn , with respect
to imbeddings Q 7! QX. Equivalently, is the quotient of R by the
principal left ideal generated by X \Gamma 1. Linear functionals on R1 are identified with
linear functionals ' on R which satisfy '(fX). The ring R1 has a similar
r-ole for the Young-Fibonacci lattice and the Okada algebra F as the ring of symmetric
functions has for the Young lattice and the group algebra of the infinite symmetric
group S1 (see [M]).
Non-commutative Jacobi determinants.
The following definition is based on a remark which appeared in the preprint version
of [Ok]. We consider two non-commutative n-th order determinants
and
By definition, the non-commutative determinant is the expression
w2Sn
sign(w) a w(1)1 a
In other words, the k-th factor of every term is taken out of the k-th column. Note that
polynomials (5.3), (5.4) are homogeneous elements of R, deg
Following Okada, we define Okada-Schur polynomials (or s-functions) as the products
(cf. [Ok], Proposition 3.5). The polynomials s v for are homogeneous of degree
n, and form a basis of the linear space Rn . We define a scalar product on the
space R by declaring s-basis to be orthonormal.
The branching of Okada-Schur functions.
We will use the formulae
obtained by decomposing the determinants (5.3), (5.4) along the last column. The first
identity is also true for case of the second identity
(5.8) can be written in the form
One can think of (5.9) as of a commutation rule for passing X over a factor of type Q 0 .
It is clear from (5.9) that
It will be convenient to rewrite (5.7), (5.8) in a form similar to (5.9):
The following formulae are direct consequences of (5.10) - (5.12):
It is understood in (5.13), (5.14) that n - 1.
The formulae (5.10), (5.13) and (5.14) imply
Theorem 5.1 (Okada). For every w 2 YFn the product of the Okada-Schur determinant
s w by X from the right hand side can be written as
This theorem says that the branching of Okada s-functions reproduces the branching
law for the Young-Fibonacci lattice. In the following statement, U is the "creation
operator" on Fun(YF), which is defined in the Appendix, (A.1.1).
Corollary 5.2. The assignment \Theta : v 7! s v extends to a linear isomorphism
taking Fun(YFn ) to Rn and satisfying \Theta ffi
Because of this, we will sometimes write U(f) instead of fX for f 2 R, and D(f)
for \Theta ffi D ffi \Theta \Gamma1 (f ), see (A.1.2) for the definition of D.
Corollary 5.3. There exist one-to-one correspondences between:
(a) Non-negative, normalized harmonic functions on YF;
(b) Linear functionals ' on Fun(YF) satisfying
(c) Linear functionals ' on R satisfying
(d) Linear functionals on
s v denotes the image of s v in
Traces of the Okada algebra F1 .
We refer to linear functionals ' on R satisfying '(s v positive linear functionals

The Okada p-functions.
Following Okada [Ok], we introduce another family of homogeneous polynomials,
labelled by Fibonacci words v 2 YF,
where
One can check that fp v g jvj=n is a Q-basis of Rn . Two important properties of the
p-basis which were found by Okada are:
Since the images of pu and of p 1u in R1 are the same, we can conveniently denote the
image by p 1 1 u . The family of p v , where v ranges over 1 1 YF, is a basis of R1 .
Transition matrix from s-basis to p-basis.
We denote the transition matrix relating the two bases fpug and fsw g by X v
jvj=n
The coefficients X v
are analogous to the character matrix of the symmetric group Sn .
They were described recurrently in [Ok], Section 5, as follows:
where m(u) is defined in (6.2) below. An explicit product expressions for the X v
be given in the next section.
x6. A product formula for Okada characters
In this Section we improve Okada's description of the character matrix X v
u to obtain
the product formula (6.11) and its consequences.
Some notation.
We recall some notation from [Ok] which will be used below. Let v be a Fibonacci
word:
Then:
if the rightmost digit of v is 1, and
is the number of 1's at the left end of v.
(6.3) The rank of v, denoted jvj, is the sum of the digits of v.
then the position of the indicated 2 is jv
1). In other words, d(v) is the product of the
positions of 2's in v.
(6.7) The block ranks of v are the numbers k 0
(6.8) The inverse block ranks of v are k t
Consider a sequence - positive integers with
a word v 2 YFn -
n-splittable, if it can be written as a concatenation
Lemma 6.1. Let - be the sequence of block ranks in a Fibonacci
word u. Then
only if, the word v is -
n-splittable.
is the -
n-splitting, then
where
Proof. This is a direct consequence of Okada's recurrence relations cited in the previous
Proposition 6.2. Let u; v 2 YFn . Let be the positions of 2's in the word
u, and put r the positions of 2's in the word v. Then
Y
Y
Proof. This can also be derived directly from Okada's recurrence relations, or from the
previous lemma. Note in particular that X v
only if, d
and j; this is the case if, and only if, v does not split according to the block ranks of
u.
We define ~
u ; from Proposition 6.2 and the dimension formula (6.5),
we have the expression
Y
Y
d s
The inverse transition matrix.
According to [Ok], Proposition 5.3, the inverse formula to Equation (5.18) can be
written in the form
juj=n
pu
We will give a description of a column X v
u for a fixed v.
Lemma 6.3. Let - be the sequence of inverse block ranks n
in a word
only if, the word u is - n-splittable.
(ii) If is the - n-splitting for u, then
where
ae \Gamma1; if ffl(u
Here denotes the number of 1's at the left end of
Proof. This is another corollary of Okada's recurrence relations cited in Section 5.
M. GOODMAN AND SERGEI V. KEROV
x7. The Martin Boundary of the Young-Fibonacci Lattice
In this section, we examine certain elements of the Martin boundary of the Young-
Fibonacci lattice YF. Ultimately we will show that the harmonic functions listed here
comprise the entire Martin boundary.
It will be useful for us to evaluate normalized positive linear functionals on the ring
(corresponding to normalized positive harmonic functions on YF) on the basis fpug.
The first result in this direction is the evaluation of the Plancherel functional on these
basis elements.
Proposition 7.1. 'P (p u words u containing at least one 2.
Proof. It follows from the definition of the Plancherel harmonic function 'P that for
v:v%w
Therefore, for all f 2 Rn ,
If
since Dp
For each word w 2 YFn , the path (w; 1w; 1 clearly satisfies the conditions of
Proposition 4.1, and therefore there is a type I harmonic function on YF defined by
Proposition 7.2. Let w 2 YFn , and let d be the positions of 2's in w. Let u
be a word in 1 1 YF containing at least one 2, and let be the positions of
2's in u. Then:
Y
Y
Proof. Let
Thus the result follows from Equation (6.12).
Next we describe some harmonic functions which arise from summable infinite words.
Given a summable infinite word w, define a linear functional on the ring R1 by the
requirements 'w
Y
Y
where are the positions of 2's in u, and the d j 's are the
positions of 2's in w. It is evident that 'w (p u that 'w is in
fact a functional on R1 .
Proposition 7.3. If w is a summable infinite Fibonacci word, then 'w is a normalized
positive linear functional on R1 , so corresponds to a normalized positive harmonic
function on YF.
Proof. Only the positivity needs to be verified. Let wn be the finite word consisting
of the rightmost n digits of w. It follows from the product formula for the normalized
characters /wn that 'w (p u
/wn (p u ). Therefore also 'w (s v
Given a summable infinite Fibonacci word w and 0 - fi - 1, we can define the
harmonic function ' fi;w by contraction of 'w , namely, '
For denote the essential rank of u, namely
ffi is the position of the leftmost 2 in u, and
Proposition 7.4. Let w be a summable infinite word and 0 - fi - 1. Let u
Then
Proof. The case
any linear functional ' on R1 , one has '(pu
A w
jxj=n
In particular,
A w
jvj=n
Note that UA w
It follows from the definitions of ' fi;w (cf.
(4.4)) and of A w
k;n that
A w
22 FREDERICK M. GOODMAN AND SERGEI V. KEROV
for f in Rn , where h\Delta; \Deltai denotes the inner product on R with respect to which the Okada
s-functions form an orthonormal basis. Recall that the operators U and D are conjugate
with respect to this inner product. Consequently,
A w
since hUA w
in Rn .
Corollary 7.5. The functionals ' fi;w for fi ? 0 and w summable are pairwise distinct,
and different from the Plancherel functional 'P .
Proof. Suppose that w is a summable word and that A is the set of positions of 2's in
w. We set
Y
Then for each k - 2 and fi ? 0,
is zero if and only if k 2 A. In particular ' fi;w 6= 'P , by Lemma 7.1, and moreover,
A n f1g is determined by the sequence of values ' fi;w (p 21 k\Gamma2 2. It is also clear
that ' fi;w (p 2 negative hence the set A and therefore also w are
determined by the values ' fi;w (p 21 k Finally, fi is determined by
for any k 62 A.
Proposition 7.6. For each summable infinite Fibonacci word w and each fi, 0 - fi - 1,
there exists a sequence v (n) of finite Fibonacci words such that '
Proof. If choose the
sequence r n so that
lim
Then, in every case,
lim
Let wn be the finite word consisting of the rightmost n digits of w, put s
and
Fix j. Suppose that u 0 has 2's at positions
1. Using the product formula for / v (n) , one
obtains
The first factor converges to 'w (p u ), so it suffices, by Proposition 7.4, to show that the
second factor converges to fi k . The second factor reduces to
Using the well-known fact that
lim
one obtains that the ratio of gamma functions is asymptotic to
which, by our choice of r n , converges to fi k , as desired.
This proposition shows that 'P as well as all of the harmonic functions ' fi;w are contained
in the Martin boundary of the Young-Fibonacci lattice. In the following sections,
we will show that these harmonic functions make up the entire Martin boundary.
x8. Regularity conditions
In this Section we obtain a simple criterion for a sequence of characters of finite dimensional
Okada algebras to converge to a character of the limiting infinite dimensional
. Using this criterion, the regularity conditions, we show that the
harmonic functions provided by the formulae (7.3) and (7.4) make up the entire Martin
boundary of the Young-Fibonacci graph YF. Technically, it is more convenient to work
with linear functionals on the spaces Rn and their limits in
with traces on F . As it was already explained in Section 5, there is a natural one-to-one
correspondence between traces of Okada algebra Fn , and positive linear functionals on
the space Rn .
In this Section we shall use the following elementary inequalities:
d
for every pair of positive integer numbers d - 2 and k;
d
and
d
d
for every pair of integers d ? k. We omit the straightforward proofs of these inequalities.
Convergence to the Plancherel measure.
We first examine the important particular case of convergence to the Plancherel
character 'P .
We define the function - of a finite or summable word v by
Y
where the d j are the positions of the 2's in v. We also recall that for each k - 2 the
function - k was defined as
Y
Note that if is a summable word, then ' v (p u according
to Equation (7.3).
Proposition 8.1. The following properties of a sequence wn 2 YF, are
equivalent:
(i) The normalized characters /wn converge to the Plancherel character, i.e.,
lim
/wn (p u
for each
(ii) lim
for every
(iii) lim
-(wn
The proof is based on the following lemmas.
Lemma 8.2. For every finite word v 2 YF, and for every of essential rank
Proof. Let indicate the positions of 2's in the word u, and let d
the positions of 2's in v. The essential rank of u can be written as
so that
Y
By the product formula,
Y
since none of the factors in the product exceed 1. In fact, j1
Lemma 8.3. For each and for every word v 2 YF,
Proof. It follows from (8.1) that
Y
Y
Y
Y
2, the last product can be estimated as
Y
and the lemma follows.
26 FREDERICK M. GOODMAN AND SERGEI V. KEROV
Lemma 8.4. If d 1 (v) 6= 2, then
Proof. We apply the inequality (8.2). If d 1 (v) - 3, then
Y
Y
by the inequality (8.2). If d 1 \Gamma1, and since d 2 - 3, the
inequality holds in this case as well.
In case of d 1
so that the second inequality of Lemma also follows from (8.2).
Proof of Proposition 8.1. The implication (i) ) (ii) is trivial, since - k
a particular character value for . The statement (iii) follows from (ii) by
Lemma 8.4. In fact, we can split the initial sequence fwng into two subsequences, fw 0
and fw 00
n g, in such a way that d 1 (w 0
2. Then we derive from Lemma
8.4 that for both subsequences -(wn
(ii) follows from (iii) by Lemma 8.3, and (ii) implies (i) by Lemma 8.2.
General regularity conditions.
We now find the conditions for a sequence of linear functionals on the spaces Rn to
converge to a functional on the limiting space
Definition (Regularity of character sequences). Let /n be a linear functional on the
graded component Rn of the ring for each assume that
the sequence converges pointwise to a functional ' on the ring R, in the sense that
for every m 2 N and every polynomial P 2 Rm . We call such a sequence regular.
Our goal in this Section is to characterize the set of regular sequences.
Definition (Convergence of words). Let fwng be a sequence of finite Fibonacci words,
and assume that the ranks jwn j tend to infinity as n !1. We say that fwng converges
to an infinite word w, iff the mth letter wn (m) of wn coincides with the mth letter w(m)
of the limiting word w for almost all n (i.e., for all but finitely many n's), and for all m.
Let us recall that an infinite word w with 2's at positions d
and only if, the series
equivalently, if the product
Y
converges.
Consider a sequence w words converging to a summable infinite
word w. We denote by w 0
n the longest initial (rightmost) subword of wn identical with
the corresponding segment of w, and we call it stable part of wn . The remaining part
of wn will be denoted by w 00
n , and referred to as transient part of wn .
Definition (Regularity Conditions). We say that a sequence of Fibonacci words wn 2
YFn satisfies regularity conditions, if either one of the following two conditions holds:
(i) lim
-(wn
(ii) the sequence wn converges to a summable infinite word w, and a strictly positive
limit
exists.
Theorem 8.5. Assume that the regularity conditions hold for a sequence wn 2 YFn .
Then the character sequence /wn is regular. If the regularity condition (i) holds, then
lim
/wn (QX n\Gammam
and if regularity condition (ii) holds, then
/wn (QX n\Gammam
for every polynomial Q Conversely, if the character sequence /wn
is regular, then the regularity conditions hold for the sequence wn 2 YFn .
This theorem will follow from Proposition 8.1 and the following proposition:
Proposition 8.6. Assume that a sequence w words converges to
a summable infinite word w, and that there exists a limit
-(wn
Then
for every generally,
/wn (p u
28 FREDERICK M. GOODMAN AND SERGEI V. KEROV
for every element u of essential rank
Proof. Let mn be the length of the stable part of the word wn , and note that mn !1.
In the following ratio, the factors corresponding to 2's in the stable part of wn cancel
out,
-(wn
Y
and a similar formula holds for the functional - k ,
Y
Consider the ratio
Y
The second product in the right hand side is a tail of the converging infinite product
(since the word w is summable), hence converges to 1, as n ! 1. By 8.3, the first
product can also be estimated by a tail of a converging infinite product,
Y
Y
j:d?mn
hence converges to 1, as well.
The proof of the formula (8.10) is only different in the way that the ratio in the left
hand side of (8.11) should be replaced by /wn (p u )='w (p u ).
Proof of the Theorem 8.5. It follows directly from Propositions 8.1 and 8.6 that the
regularity conditions for a sequence w imply convergence of functionals /wn to
the Plancherel character 'P in case (i), and to the character ' fi;w in case (ii). By the
Corollary 7.5 we know that the functions ' fi;w are pairwise distinct, and also different
from the Plancherel functional 'P .
Let us now assume that the sequence /wn converges to a limiting functional '. We
can choose a subsequence /wnm in such a way that the corresponding sequence wmn
converges digitwise to an infinite word w. If w is not summable, then
with the Plancherel functional, and the part (i) of the regularity conditions holds. Oth-
erwise, we can also assume that the limit (8.6) exists, and hence
8.6. Since the parameter fi and the word w can be restored, by Corollary 7.5, from the
limiting functional ', the sequence wn cannot have subsequences converging to different
limits, nor can the sequence -(wn ) have subsequences converging to different limits. It
follows, that the regularity conditions are necessary. The Theorem is proved.
In the following
statement,\Omega refers to the space defined at the end of Section 3.
Theorem 8.7. The map
is a homeomorphism
of\Omega onto the Martin boundary of YF.
Proof. It follows from Corollary 7.5 that the map is an injection
of\Omega into the Martin
boundary, and from Theorem 8.5 that the map is surjective. Furthermore, the proof
that the map is a homeomorphism is a straightforward variation of the proof of the
regularity statement, Theorem 8.5.
x9. Concluding remarks
The Young-Fibonacci lattice, along with the Young lattice, are the most interesting
examples of differential posets. There is a considerable similarity between the two
graphs, as well as a few severe distinctions.
Both lattices arise as Bratteli diagrams of increasing families of finite dimensional
semisimple matrix algebras, i.e., group algebras of symmetric groups in case of Young
lattice, and Okada algebras in case of Young-Fibonacci graph. For every Bratteli dia-
gram, there is a problem of describing the traces of the corresponding inductive limit
algebra, which is well-known to be intimately related to the Martin boundary construction
for the graph. The relevant fact is that indecomposable positive harmonic functions,
which are in one-to-one correspondence with the indecomposable traces, form a part of
the Martin boundary.
For the Young lattice the Martin boundary has been known for several decades,
and all of the harmonic functions in the boundary are known to be indecomposable
(extreme points). In this paper we have found the Martin boundary for the Young-
Fibonacci lattice. Unfortunately, we still do not know which harmonic functions in the
boundary are decomposable (if any). The method employed to prove indecomposability
of the elements of the Martin boundary of the Young lattice can not be applied to
Young-Fibonacci lattice, since the K 0 -functor ring R of the limiting Okada algebra F
is not commutative, as it is in case of the group algebra of the infinite symmetric group
(in this case it can be identified with the symmetric function ring).
Another natural problem related to Okada algebras is to find all non-negative Markov
traces. We plan to address this problem in another paper.


Appendix


In this appendix, we survey a few properties of differential posets introduced by R.
Stanley in [St1-3]. A more general class of posets had been defined by S. Fomin in
[F1-3]. In his terms differential posets are called self - dual graphs.
A.1 Definitions.
A graded poset
called branching diagram (cf. [KV]), if
(B1) The set \Gamma n of elements of rank n is finite for all
(B2) There is a unique minimal element
There are no maximal elements in \Gamma.
One can consider a branching diagram as an extended phase space of a non - stationary
being the set of admissible states at the moment n and covering
relations indicating the possible transitions.
We denote the rank of a vertex the number of saturated
chains in an interval [u; v] ae \Gamma by d(u; v).
Following [St1], we define an r-differential poset as a branching diagram \Gamma satisfying
two conditions:
(D1) If u 6= v in \Gamma then the number of elements covered by u and v is the same as
the number of elements covering both u and v
exactly k elements, then v is covered by exactly
of \Gamma.
Note that the number of elements in a differential poset covering two distinct elements
can be at most 1. In this paper we focus on 1-differential posets.
For any branching diagram \Gamma one can define two linear operators in the vector space
Fun(\Gamma) of functions on \Gamma with coefficients in Q: the creation operator
w:v%w
and the annihilation operator
u:u%v
finitely supported functions on \Gamma with formal linear combinations of points
of \Gamma and vertices of \Gamma with the delta functions at the vertices, one can write instead:
w:v%w
u:u%v
u:
One can characterize r-differential posets as branching diagrams for which the operators
U; D satisfy the Weyl identity DU \Gamma
A.2 Some properties of differential posets.
We review below only a few identities we need in the main part of the paper. For a
general algebraic theory of differential posets see [St1-3], [Fo1-3]. Assume here that \Gamma
is a 1-differential poset.
The first formula is well known:
w:v%w
Proof. Let
can be written
as . This is trivial for assuming D
Our next result is a generalization of (A.2.1).
Lemma A.2.2. Let \Gamma be a 1-differential poset, and let u - v be any vertices of ranks
w:v%w
Proof. Using the notation dn
jvj=n d(u; v) v, one can easily see that U dn
dn+1 (u) and that (A.2.3) can be rewritten in the form
For the formula is true by the definition of D. By the induction argument,
Note that (A.2.3) specializes to (A.2.1) in case
A.3 Plancherel transition probabilities on a differential poset.
It follows from (A.2.1) that the numbers
can be considered as transition probabilities of a Markov chain on \Gamma. Generalizing the
terminology used in the particular example of Young lattice (see [KV]), we call (A.3.1)
Plancherel transition probabilities.
Lemma A.3.2. Let u - v be the vertices of ranks in a 1-differential
poset \Gamma. Then the Plancherel probability p(u; v) to reach (by any path) the vertex v
starting with u is
Proof. We have to check that
d(u; w), we obtain
jvj=n
and the Lemma follows.



--R


Dimensions and C
Generalized Robinson - Schensted - Knuth correspondence
Duality of graded graphs
Schensted algorithms for dual graded graphs
Algebras associated to the Young-Fibonacci lattice

Variations on

Further combinatorial properties of two Fibonacci lattices
Discrete potential theory and boundaries
The Grothendieck Group of the Infinite Symmetric Group and Symmetric Functions with the Elements of the K 0

2nd edition
Asymptotic character theory of the symmetric group
Thoma's theorem and representations of infinite bisymmetric group
Analysis and its Appl.
The Boundary of Young Graph with Jack Edge Multiplicities
Department of Mathematics
--TR
Further combinatorial properties of two Fibonacci lattices
Schensted Algorithms for Dual Graded Graphs

--CTR
Jonathan David Farley , Sungsoon Kim, The Automorphism Group of the Fibonacci Poset: A Not Too Difficult Problem of Stanley from 1988, Journal of Algebraic Combinatorics: An International Journal, v.19 n.2, p.197-204, March 2004
differential poset;okada algebra;harmonic function;non-commutative symmetric function;martin boundary