\magnification=1200
\input wizzle
\input mcfile
\input labelfig
\input newinsert
\input epsf
\tolerance 1000
\def\oddmargin{1pt}
%\voffset -0.5truein
\hoffset 5truecm
\hfuzz 6pt
\baselineskip 12pt

\font\smcp=cmcsc8 
\headline={\ifnum\pageno>1 {\smcp the electronic journal of combinatorics 4 (1997),
        \#R22\hfill\folio} \fi} 

\font\ams=msbm10
\def\nnleq{\hbox{\ams\char'000}}
\def\nngeq{\hbox{\ams\char'001}}


\font\ext=cmex10
\def\exline{\hbox{\ext\char'054}}


%\font\sym=cmsy10 scaled\magstep 5
%\def\dsim{\hbox{\sym\char'031}}

\font\bold=cmmib10
\def\kb{\hbox{\bold\char'153}}
\def\lb{\hbox{\bold\char'154}}
\vsize 9 true in

\ctr{\twelveb Asymptotics of Young Diagrams and Hook Numbers  }
\ske
\ctr{{\bf Amitai Regev}\footnote{$^*$}{Work partially supported by N.S.F.
Grant No.DMS-94-01197.}}
\inst
\par
\ctr{and}
\par
\ctr{\DM}
\ctr{The Pennsylvania State University}
\ctr{University Park, PA 16802,\ \  U.S.A. }
\ske
\ctr{{\bf Anatoly Vershik}\footnote{$^\dagger$}{Partially supported by
Grant INTAS 94-3420 and Russian Fund 96-01-00676   }}
\par
\ctr{St. Petersburg branch of the Mathematics Institute}
\ctr{of the Russian Academy of Science }
\ctr{ Fontanka 27}
\ctr{St. Petersburg, 191011  Russia}
\par
\ctr{and}
 \par
\ctr{The Institute for Advanced Studies of the Hebrew University }
\ctr{Givat Ram }
\ctr{Jerusalem, Israel}
%
\baselineskip 15pt
%
\vskip .2in
\centerline{Submitted: August 22, 1997; Accepted: September 21, 1997}


\vskip 1.2cm
{\bf Abstract:} \ \ Asymptotic calculations are applied to study 
the degrees of certain sequences of characters of symmetric groups. 
Starting with a given partition $\mu$, we deduce several skew diagrams 
which are related to $\mu$. To each such skew diagram there corresponds 
the product of its hook numbers. By asymptotic methods we obtain some 
unexpected arithmetic properties between these products.  The authors do 
not know "finite", nonasymptotic proofs of these results. The problem 
appeared in the study of the hook formula for various kinds of Young 
diagrams. The proofs are based on properties of shifted Schur functions, 
due to Okounkov and Olshanski. The theory of these functions arose from 
the asymptotic theory of Vershik and Kerov of the representations of the 
symmetric groups.


\endpage
\baselineskip 18pt
%
{\bf \S1.\ \ Introduction and the main results}
\par
Asymptotic calculations are applied to study the degrees of certain
sequences of \ch s of symmetric groups $S_n,\  n\ra\nty$. We obtain
some unexpected arithmetic properties of the set of the hook numbers
for some special families of (fixed) skew-Young diagrams (Theorem 1.2).
The problem appeared in the study of the hook formula for various kinds
of Young diagrams. The proof of 1.2
is based on the properties of shifted Schur
functions \(Ok.Ol\) which appeared in the asymptotic theory of the
\rept\ of the symmetric groups in \(Ver.Ker\). The authors do not know
a ``finite" proof of the theorem.
\par
Given a partition $\mu$, we describe in 1.1 - a construction of certain
skew diagrams which are derived from  $\mu$: these are $SQ(\mu),\ SR(\mu)
,\ SR(\mu\pr),\ R$ and $D_\mu$ below. Next, one fills these skew diagrams
with their \corr ing hook numbers \(Mac, page 10\). Theorem 1.2, which is
the main result here, gives some divisibility properties of
the products of these hook numbers.
\par
We remark again that even though the statement of theorem 1.2 has
nothing to do with asymptotics, its proof does use asymptotic methods.
It should be interesting to find an ``asymptotic free" proof of theorem
1.2.
\par
We start with
\par
{\bf 1.1:\ \  A Construction:} \ \ Given a partition (= diagram) $\mu$,
let $D^*_\mu$ denote the double reflection of $\mu$. For example, if
$\mu=(4,2,1)$ then
%
$$ D_\mu=\matrix{x & x & x & x \cr x & x & & \cr x & & & \cr}\ \qquad \
and \ \qquad
D^*_\mu = \matrix{& &  & x \cr & & x & x \cr x & x & x & x \cr}. $$
\ske\noindent
Recall that $\mu\pr_1=\ell(\mu)$ is the number of nonzero parts of
$\mu$. Complete $D_\mu^*$  to the $\mu_1\times\mu\pr_1$
rectangle $R(\mu)$, then  draw $D^*_\mu$ on top and on the left of $R$.
Finally, erase the first $D^*_\mu$. Denote the resulting skew diagram by
$SQ(\mu)$. For example, with $\mu=(4,2,1)$ we get\vskip 0.5truecm
%
%REGFIG.ONE
%
\hbox{\baselineskip 12pt
\def\vl{\vrule height 10pt depth 8pt}  %20 & 10
\def\hl{\leaders \hrule height 3pt depth -2.5pt\hfill}
%
$$ \vbox{\halign{\hskip 50pt
$\lft{#}$ & \lft{#} \qquad& $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$  & $\lft{#}$ & $\lft{#}$
\hquad & $\lft{#}$\hquad & $\lft{#}$\hquad & $\lft{#}$\cr
%
&& && && && & A_2\cr
&& && && && && \searrow &&x\cr
&&&&&&&&&&& x & x \cr
&& && && && & x & x & x & x \cr
\noalign{\vskip -6pt}
\noalign{\hskip 194.7pt{\hbox to 78pt{\hl}}}
\noalign{\vskip -5pt}
SQ(4,2,1)&=&&& A_1 &&&&\vl\cr
\noalign{\vskip -2pt}
&&&&& \searrow &&x&\vl & x & x & x\cr
\noalign{\vskip -2pt}
& & &&& & x & x & \vl & x & x \cr
\noalign{\vskip -6pt}
& & & &&&  &  & \vl &  &  &\nwarrow\cr
\noalign{\vskip -12pt}
&&& & x & x & x & x & \vl\cr
\noalign{\vskip -12pt}
& &&&  &  &  &  & \vl &&&& A\cr
}} $$}\vskip 0.5truecm
%
We subdivide $SQ(\mu)$ into the three areas $A,\ A_1$ and $A_2$: $A=R-D^*
_\mu,\ A_1$ is the $D^*_\mu$ on the left of $R$ and $A_2$ is the $D^*_
\mu$ on top of $R$. Denote $SR(\mu)=A_1\cup A$, the ``shifted rectangle".
\par
Clearly, $\v A\cup A_1\v=\v A\cup A_2\v=\v R\v,\ \v A_1\v=\v A_2\v=\v
\mu\v$, so $\v SQ(\mu)\v=\v R\v+\v\mu\v$. Now, fill $SQ(\mu),SR(\mu),\
R$ and $\mu$ with their hook numbers. For example, when $\mu=(4,2,1)$
\ske
%
\hbox{\baselineskip 12pt
\def\vl{\vrule height 10pt depth 8pt}  %20 & 10
\def\hl{\leaders \hrule height 3pt depth -2.5pt\hfill}
%
$$ \vbox{\halign{\hskip 50pt
$\lft{#}$ & \lft{#} \qquad& $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$ \hquad & $\lft{#}$ \hquad & $\lft{#}$ \hquad &
$\lft{#}$  & $\lft{#}$ & $\lft{#}$
\hquad & $\lft{#}$\hquad & $\lft{#}$\hquad & $\lft{#}$\cr
%
&& && && && && && && 3 \cr
&& && && && && &&  4 && 2 \cr
&& && &&  &&  & 6 && 5 & 3 && 1 \cr
\noalign{\vskip -6pt}
\noalign{\hskip 180.7pt{\hbox to 70pt{\hl}}}
\noalign{\vskip -5pt}
SQ(4,2,1)& : && &  && &&\vl    \cr
\noalign{\vskip -2pt}
&& && & &&  6 &\vl & 4 && 3 & 1\cr
\noalign{\vskip -2pt}
& & && & & 5 & 4 & \vl & 2 && 1 \cr
\noalign{\vskip -6pt}
& & & &&&  &  & \vl &  &  &\cr
\noalign{\vskip -12pt}
&&& & 4 & 3 & 2 & 1 & \vl\cr
\noalign{\vskip -12pt}
& &&&  &  &  &  & \vl &&&& \cr
}} $$}\vskip 0.5truecm
%
$$ SR(4,2,1):\hskip 2cm \matrix{& & & 6 & 4 & 3 & 1 \cr
& & 5 & 4 & 2 & 1 \cr   4 & 3 & 2 & 1 & & & \cr} $$
%
\ske
$$ R(4,2,1):\hskip 2cm  \matrix{6 && 5 && 4 && 3 \cr 5 && 4 && 3 && 2 \cr
4 && 3 && 2 && 1 \cr} $$
\ske
and
$$ (4,2,1):\hskip 2cm \matrix{6 && 4 && 2 && 1 \cr 3 && 1 && && \cr
1 && && && \cr} $$
\ske
Thus, for example, $\prod_{x\in(4,2,1)}h(x)=1^3\cdot 2\cdot 3\cdot 4
\cdot 6=144$.
\par
Note that the hook numbers in $SR(\mu)$ are the same as those in the
area $A_1\cup A$ of $SQ(\mu)$.
\par
As usual, $\mu\pr_1=\ell(\mu)$ is the number of nonzero parts of $\mu$.
Recall that $s_\mu(x_1,x_2,\cdots)$ is the \corr ing Schur function,
and $s_\mu\underbrace{(1,\cdots,1)}_{\mu\pr_1}$ is the number of (semi%
-standard, i.e. rows weakly and column strictly increasing) tableaux of
shape $\mu$, filled with elements from $\{1,2,\cdots,\mu\pr_1\}$ \(Mac\).
Similarly for $s_{\mu\pr}\underbrace{(1,\cdots,1)}_{\mu_1}$.
\par
{\bf 1.2 \ \ Theorem:} \ \ Let $\mu$ be a partition. With the above
construction of $SQ(\mu)=A\cup A_1\cup A_2$ and $R$, we have
$$ \left(\prod_{x\in R}\ h(x)\right)\raise 15pt\hbox{${\exline}$}
\left(\prod_{x\in A_1\cup A}\ h(x)\right)=s_\mu  (\underbrace{1,\cdots,
1}_{\mu\pr_1}). \ \ \leqno(1) $$
%
\Ip, $\prod_{x\in A_1\cup
A}h(x)$ divides $\prod_{x\in R}h(x)$. \(Note that $A\cup A_1\st SQ(\mu)$,
and for $x\in A_1\cup A,\ h(x)$ is the \corr ing hook number in $x\in
SQ(\mu)$\).
%
\parno
(1')\ \ Similarly,
$$ \left(\prod_{x\in R}\ h(x)\right)\raise 15pt\hbox{${\exline}$}
\left(\prod_{x\in A_2\cup A}\
h(x)\right)=s_{\mu\pr}(\underbrace{1,\cdots,1}_{\mu_1}). $$
\par
$$\prod_{x\in SQ(\mu)}\ h(x)=\left(\prod_{x\in R}\ h(x)\right)\cdot\left(
\prod_{x\in\mu}\ h(x)\right).\ \leqno(2) $$
\par
We conjecture that a statement much stronger than 1.2.2 holds, namely:
 the  two  multisets   \bk
$\{h(x)\mid x\in SQ(\mu)\}$ and $\{h(x)\mid x\in R\}\cup\{h(x)\mid
x\in\mu\}$ are equal.
\par
Theorem 1.2.1 is an obvious consequence of the following ``asymptotic"
theorem.
\par
{\bf 1.3.\ \ Theorem:} \ \  Let $\mu=(\mu_1,\cdots,\mu_k)$, be a
partition. Let $n=k\ell$, \bk
$\mu_1\leq\ell\ra\nty$, and denote $\l=\l(\ell)
=(\ell^k)$. Then
$$ \lim_{\ell\ra\nty}\ {d_{\l/\mu} \over d_\l} = \left({1\over k}
\right)^{\v\mu\v}\cdot s_\mu(\underbrace{1,\cdots,1}_k) \ \leqno(a) $$
and
$$\lim_{\ell\ra\nty}\ {d_{\l/\mu}\over d_\l}= \left({1
\over k}\right)^{\v\mu\v}\cdot\left(\prod_{x\in R(\mu_1,\mu\pr_1)}h(x)
\right)\raise 15pt\hbox{${\exline}$}\left(\prod_{x\in A_1\cup A}
\ h(x)\right).  \ \leqno(b) $$
\parno
Theorem 1.2.1' follows from 1.2.1 by conjugation.
\par
Theorem 1.2.2 is a consequence of the following ``asymptotic" theorem
\par
{\bf 1.4. \ \ Theorem:}\ \ Let $\mu$ be a fixed partition. Let $\mu_1
\leq\ell\ra\nty$, \bk
$\mu\pr_1\leq m\ra\nty,\ n=\ell m$ and $\l=\l(\ell,m)=
(\ell^m)$. Then
$$ \lim_{\ell,m\ra\nty}\ {d_{\l/\mu}\over d_\l}=\
{1\over \prod_{x\in\mu}\ h(x)}\ . \ \leqno(a) $$
%
$$ \lim_{\ell,m\ra\nty}\ {d_{\l/\mu}\over d_\l}=\
\left(\prod_{x\in R}\ h(x)\right)\raise 15pt\hbox{${\exline}$}
\left(\prod_{x\in SQ(\mu)}\ h(x)\right)\ . \ \leqno(b) $$
\par
In this note we apply the following main tools:
\parno
a)\ \ The theory of symmetric functions \(Mac\). \Ip, we apply the hook
formula
$$ d_\l=\ {\v\l\v!\over\prod_{x\in\l}\ h(x)} $$
and I.3, Example 4, page 45 in \(Mac\).
\parno
b) \ \ The Okounkov-Olshanski \(Ok.Ol\) theory of ``shifted symmetric
functions". \Ip, we apply formula (0.14) of \(Ok.Ol\): \parno
Let $\mu\vdash k,\ \l\vdash n,\ k\leq n,\ \mu\st\l$, then
$$ {d_{\l/\mu}\over d_\l}\ =\ {s^*_\mu(\l)\over n(n-1)\cdots(n-k+1)}
\ . $$
\parno
Here $s^*_\mu(x)$ is the ``shifted Schur function" \(Ok.Ol\);\ \ one of
its key properties is that  \bk
$s^*_\mu(x)=s_\mu(x)+$ lower terms, where
$s_\mu(x)$ is the ordinary Schur function.
\par
We remark that the paper \(Ok.Ol\) was influenced by the work of
Vershik and Kerov on the asymptotic theory of the \rep ations of the
symmetric groups. See for example \(Ver.Ker\), in which the \ch s of
the infinite symmetric group are found from limits involving ordinary
Schur functions. See also the introduction of \(Ok.Ol\).
%
\ske
{\bf \S2.}\ \ Here we prove theorem 1.3 which, as noted before, implies
1.2.1 (and 1.2.1').
\par
{\bf 2.1. \ \ The proof of theorem 1.3.}
$$ {d_{\l(\ell)/\mu}\over d_{\l(\ell)}}\ =\ {s^*_\mu(\l_1(\ell),\cdots,
\l_k(\ell))\over n(n-1)\cdots(n-\v\mu\v+1)}\ , $$
where $n=\v\l\v=k\ell$. Since $\ell\ra\nty,\ n(n-1)\cdots
(n-\v\mu\v+1)\simeq (k\ell)^{\v\mu\v}$. Also,
$$ s^*_\mu(\l)=s_\mu(\l)+(lower\ terms\ in\ \ n), $$
hence
$$ s^*_\mu(\l)\simeq s_\mu(\l)=s_\mu\underbrace{(\ell,\cdots,\ell)}_k
. $$
%
Recall that for two sequences $a_n$, $b_n$ of real numbers,
$a_n \simeq b_n$ means that $\lim_{n\rightarrow\infty} 
{a_n\over b_n} = 1$.\par
%
Since $s_\mu(x)$ is \hog\ of degree $\v\mu\v$,
$$ s_\mu(\l)=\ell^{\v\mu\v}\cdot s_\mu(\underbrace{1,\cdots,1}_k)
\ . $$
The proof now follows easily.
\hfill\QED\par
%
\ske
{\bf 2.2. \ \ The proof of theorem 1.3.b:} \ \ Since $\l$ is a
rectangle, hence $d_{\l/\mu}=d_\eta$, where $\eta$
is the double reflection of $\l/\mu$. Denote by $\tilde\mu=D^*_
\mu$  the double reflection of $\mu$. Thus
\ske
%
%REGFIG.TWO
%
\midinsert
\SetLabels
(.05*.46)$\eta$\\
(.85*.35)$D^*_\mu$\\
(.51*.81)$\lscr$\\
\endSetLabels
%\ShowGrid
\centerline{\AffixLabels{%
\epsfysize=1.9in\epsfbox{regfig2.eps}
}}
\endinsert
%
To calculate $d_\l$ and $d_\eta$ by the hook formula, fill $\l=\l(\ell)$
and $\eta$ with their respective hook numbers. In both, examine the
$i^{th}$ row from the bottom - with their respective hook numbers.
Divide $\eta$ into $B_1$ and $B_2$ as follows:
%
%REGFIG3
%
\midinsert
\SetLabels
(.68*.48)$B_1$\\
(.27*.48)$B_2$\\
(.86*.43)$D^*_\mu$\\
(.50*.85)$\lscr$\\
%
(.79*.77)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
   \hbox to 35pt{\upbracefill}
   \hbox to 35pt{\hfill$\mu_1$\hfill}}} $$\\
%
(.68*.15)
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
   \hbox to 35pt{\upbracefill}
   \hbox to 35pt{\hfill$\mu_1$\hfill}}} $$\\
%
\endSetLabels
%\ShowGrid
%\centerline
\hskip 41pt{\AffixLabels{%
\epsfysize=2.2in\epsfbox{regfig3.eps}
}}
\endinsert
%
\parno
Notice that $B_1=SR(\mu)$ of 1.1. Note also that the hook numbers in
$B_1$ are those in $SR(\mu)$, and they are \ind\ of $\ell$.
\par
Examine the hook numbers in $B_2$. In the $i^{th}$ row (from bottom),
these are  \bk
$\mu_1+i,\ \mu_1+i+1,\cdots,\ell+i-1-\mu_i$, consecutive integers.
\par
We also divide $\l(\ell)$ into two rectangles:
%
%REGFIG.FOUR
%
\midinsert
\SetLabels
%
(.86*.03)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
   \hbox to 35pt{\upbracefill}
   \hbox to 35pt{\hfill$\mu_1$\hfill}}} $$\\
%
(.03*.4)$\lambda(\lscr)$\\
(.87*.4)$R_1$\\
(.51*.81)$\lscr$\\
(.49*.4)$R_2$\\
\endSetLabels
%\ShowGrid
\centerline{\AffixLabels{%
\epsfysize=1.9in\epsfbox{regfig4.eps}
}}
\endinsert
%
%
\parno
Again, the hook numbers in $R_1$ are \ind\ of $\ell$, and those in the
$i^{th}$ row (from bottom) of $R_2$ are $\mu_1+i,\mu_1+i+1,\cdots,\ell+i
-1$, again  consecutive integers.
\par
By the ``hook" formula, the left hand side of 1.3.b is
$$ {d_{\l(\ell)/\mu}\over d_{\l(\ell)}}\ =\ {d_\eta\over d_{\l(\ell)}}\
=\ \left\({(n-\v\mu\v)!\over \prod_{x\in\eta}h(x)}\right\)
 \raise 15pt\hbox{${\exline}$}  \left\(
{n!\over\prod_{x\in\l(\ell)}\ h(x)}\right\) $$
$$ =\ {(n-\v\mu\v)!\over n!}\ \cdot\ \left\({\prod_{x\in\l(\ell)}h(x)
\over \prod_{x\in\eta}h(x)}\right\) $$
%
where $n=k\ell$. Since $\ell\ra\nty$,
$$ {(n-\v\mu\v)!\over n!}\ \simeq\ \left({1\over n}\right)
^{\v\mu\v}=\ \left({1\over k\ell}\right)^{\v\mu\v}  . $$
\parno
Now
$${\prod_{x\in\l(\ell)}h(x)\over\prod_{x\in\eta}h(x)}\ = \ \left\({\prod
_{x\in R_1}h(x)\over \prod_{x\in B_1}h(x)}\right\)\cdot\left\({\prod_{
x\in R_2}h(x)\over \prod_{x\in B_2}h(x)}\right\)=\a\cdot\b. $$
\parno
Note that the right hand side of 1.3.b is $({1\over k})^{\v\mu\v}\cdot\a$.
\par
We calculate $\b$:
$$ \prod_{x\in R_2}h(x)=\prod^{\mu_1\pr}_{i=1}\((\mu_1+i)(\mu_1+i+1)\cdots
(\ell+i-1)\), $$
$$ \prod_{x\in B_2}h(x)=\prod^{\mu\pr_1}_{i=1}\((\mu_1+i)(\mu_1+i+1)\cdots
(\ell+i-1-\mu_i)\), $$
thus
$$ \b=\prod^{\mu\pr_1}_{i=1}\((\ell+i-\mu_i)(\ell+i-\mu_i+1)\cdots(\ell+
i-1)\)\simeq\ell^{\v\mu\v}, $$
(since $\ell\ra\nty$).
\par
Hence,
$$ \lim_{\ell\ra\nty}\ {d_{\l(\ell)/\mu}\over d_{\l(\ell)}}\ =\
\left({1\over k}\right)^{\v\mu\v}\cdot\a $$
and the proof is complete.
\hfill\QED\par
\ske
%
{\bf \S3.}\ \ Here we prove theorem 1.4 which, as noted before, implies
theorem 1.2.2.
\par
{\bf 3.1. \ \ The proof of 1.4.a:} \ \ Let $\l=\l(\ell,m)=(\ell^m),\
\ell, m\ra\nty$. We show first that $s^*_\mu(\l)\simeq s_\mu(\l)$, as
follows:\ \ By \(Ok.Ol.(0.9)\),
$$ \eqalign{e^*_r(\l) & =\sum_{i\leq i_1<\cdots < i_r\leq m}\ (\ell+r-1)
(\ell+r-2)\cdots\ell =  \cr
& = (\ell+r-1)(\ell+r-2)\cdots\ell\cdot{m\choose r}\simeq{\ell^r m^r
\over r!} \ .  \cr} $$
\parno
Similarly, $e_r(\l)\simeq {\ell^r m^r\over r!}$ .
\par
Let $\emt$ be given as in \(Ok.Ol.\S13\). By \(Ok.Ol.(13.8)\) it easily
follows that for any $u$ and $r$,
$$ \emt^{-u}e^*_r(\l)\simeq e^*_r(\l)\simeq e_r(\l). $$
\parno
Applying the Jacobi Trudi formulas for $s_\mu(\l)$ |(Mac. I, (3.5),
page 41\) and for $s^*_\mu(\l)$ \(Ok.Ol.(13.10)\), it clearly follows
that $s^*_\mu(\l)\simeq s_\mu(\l)$. Now in 2.1, here
$$ {d_{\l(\ell,m)/\mu}\over d_{\l(\ell,m)}}\ = \ {s^*_\mu(\l_1(\ell,m),
\cdots,\l_{m+k}(\ell,m))\over n(n-1)\cdots (n-\v\mu\v+1)} $$
where
$$ n= \ell m.$$
Here
$$ s^*_\mu(\l(\ell,m))\simeq s_\mu(\l(\ell,m)) = \ell^{\v\mu\v}
s_\mu (\underbrace{1,\cdots,1}_m). $$
Thus
%
$$ {d_{\l(\ell,m)/\mu}\over d_{\l(\ell,m)}}\ \simeq\ \left({1\over n}
\right)^{\v\mu\v}\cdot s_\mu (\underbrace{1,\cdots,1}_m) =
\left({1\over m}\right)^{\v\mu\v}\cdot\prod_{x\in\mu}\ {m+c(x)
\over h(x)}\ , $$ 
%
(\(Mac, pg. 45, Ex 4\))
%
where $c(x)$ is the content of $x\in\mu$. Since $m\ra\nty,\ \ m+c(x)
\simeq m$ for all $x\in\mu$, and the proof follows.
\hfill\QED\par
%
\ske
{\bf 3.2.\ \ The proof of 1.4.b:} \ \ Choose $\ell,m$ large so that $\mu
\st\l(\ell,m)$. Let $\eta$ be the double reflection of $\l(\ell,m)/\mu$,
so $d_{\l(\ell,m)/\mu}=d_\eta$,
then calculate $d_\eta$ by the hook formula.
To analyze the hook numbers in $\eta$, we subdivide $\eta$ into the
areas $A_{1,\eta},\cdots,A_{4,\eta}$ as shown below:
%
%REGFIG.SIX
%
\midinsert
\SetLabels
(-.02*.43)$\eta:$\\
(.05*.425)$m$\\
(.41*.71)$\lscr$\\
(.35*.48)$A_{1,\eta}$\\
(.7*.48)$A_{3,\eta}$\\
(.35*.22)$A_{2,\eta}$\\
(.82*.33)$A_{4,\eta}$\\
(.8*.1)$D^*_\mu$\\
(.78*.215)$\big\}\mu^\prime_1$\\
%
(.685*.135)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
   \hbox to 5pt{\upbracefill}
   \hbox to 5pt{\hfill$\mu_1$\hfill}}} $$\\
%
\endSetLabels
%\ShowGrid
%\centerline
\hskip 25pt{\AffixLabels{%
\epsfysize=2.7in\epsfbox{regfig6.eps}
}}
\endinsert
%
\parno
i.e., $D^*_\mu$ is drawn at the bottom-right of the $\ell\times m$
rectangle. We then follow 1.1 and construct $A_{4,\eta}=SQ(\mu)$. Now
$A_{1,\eta}$ is the $(\ell-\mu_1)\times(m-\mu\pr_1)$ rectangle, and this
determines $A_{2,\eta}$ and $A_{3,\eta}$.
\par
We also split the $\ell\times m$ rectangle $\l=\l(\ell,m)$ accordingly:
%
%REGFIG.SEVEN
%
\midinsert
\SetLabels
%
(.38*.58)$A_{1,\lambda(\lscr,m)}$\\
(.38*.23)$A_{2,\lambda(\lscr,m)}$\\
(.89*.55)$A_{3,\lambda(\lscr,m)}$\\
(.89*.08)$A_{4,\lambda(\lscr,m)}$\\
(.83*.22)$\bigg\}\mu^\prime_1$\\
%
(.75*.12)%
$$ \hbox{\vtop to 14pt{\baselineskip 8pt
   \hbox to 25pt{\upbracefill}
   \hbox to 25pt{\hfill$\mu_1$\hfill}}} $$\\
%
\endSetLabels
%\ShowGrid
\centerline{\AffixLabels{%
\epsfysize=1.9in\epsfbox{regfig7.eps}
}}
\endinsert
%
%
\vskip 1cm
Since $\l(\ell,m)\vdash\ell m$ and $\eta\vdash\ell m-\v\mu\v$,
$$ {d_\eta\over d_{\l(\ell,m)}}\simeq\left({1\over \ell m}\right)^{\v\mu
\v}\cdot{\prod_{x\in\l(\ell,m)} h_{\l(\ell,m)}(x)\over \prod_{x\in\eta}
h_\eta(x)}. $$
\par
Now, $h_{\l(\ell,m)}(x)=h_\eta(x)$ for $x\in A_{1,\eta}=A_{1,\l
(\ell,m)}$.  As in 2.3
$$ {\prod_{x\in A_{2,\l(\ell,m)}}h_{\l(\ell,m)}(x)\over\prod_{x\in
A_{2,\eta}}h_\eta(x)}\ \simeq\ \ell^{\v\mu\v}. $$
\parno
Similarly (or, by conjugation),
$$ {\prod_{x\in A_{3,\l}}h_{\l(\ell,m)}(x)\over\prod_{x\in A_{3,\eta}}
h_\eta(x)} \ = \ m^{\v\mu\v}\ . $$
\par
After cancellations we have
$$ {d_\eta\over d_\l}\simeq{\prod_{x\in A_{4,\l}}h_{\l(\ell,m)}(x)\over
\prod_{x\in A_{4,\eta}}h_\eta(x)}\ = \ {\prod_{x\in R(\mu_1,\mu\pr_1)}
h(x)\over\prod_{x\in SQ(\mu)}h(x)} $$
%
and the proof is complete.
\hfill\QED\par
\ske
%

\baselineskip 12pt
\ctr{\bf References}\bigskip
%
\offset{40pt}
{\(Ok.Ol\)}Okounkov A. and Olshanski G., Shifted Schur functions,\ \
preprint.\par
%
\offset{40pt}
{\(Mac\)}Macdonald I.G.,  Symmetric functions and Hall \po s, \ \
Oxford University Press, 2nd edition 1995.\par
%
\offset{45pt}
{\(Ver.Ker\)}Vershik A.M. and Kerov, S.V., Asymptotic Theory of \ch s
of the symmetric group, Funct. Anal. Appl. 15 (1981) 246-255.
%

\bk\bk
email addresses: regev@wisdom.weizmann.ac.il, vershik@pdmi.ras.ru 


\bye