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\boldmath

\LARGE\bf

%-----------------------------------------------------------
\begin{center}

\centerline{\resizebox{11.5in}{!}{\includegraphics*{MIT3.jpg}}}
\vskip-6.8in

{\huge\bf The Combinatorial Definition of
\\
\medskip
the Schur, Zonal, and Jack
Polynomials}

% of a Matrix Argument}
\bigskip

\end{center}

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{\LARGE \bf
\begin{center}
Plamen Koev \\
{\Large Massachusetts Institute of Technology }


\vspace*{.5in}

%\vspace*{1in}

% {\rm \Large \bf Joint work with Alan Edelman}

\vspace*{1in}

{\rm \Large \bf SAMSI, October 19, 2006}


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}



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% ----------------------------------------------------------

\slide{Start with the Schur functions. Part One: Partitions}

\begin{itemize}

\item Definition: Partition $\lambda$ of the integer $n$; denoted $\lambda\vdash n$

\bigskip

... is the $m$-tuple
$\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_m)$ where
$n=\lambda_1+\lambda_2+\ldots+\lambda_m$ and
$\lambda_1\geq\ldots\geq\lambda_m\geq 0$.

\item Example: Partition
\begin{eqnarray*}
(1)&\vdash& 1\\
\\
(1,1)&\vdash& 2\\
(2)&\vdash &2\\
\\
(1,1,1)&\vdash &3\\
(2,1)&\vdash &3\\
(3)&\vdash &3\\
\\
\cdots\\
(4,2,1)&\vdash &7
\end{eqnarray*}

% ---------------------------------------------------------------

\slide{Schur functions. Part Two: Young Diagrams}

\item Definition: The Young diagram of the partition $(4,2,1)\vdash 7$ is

\bigskip
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\centerline{
\vbox{\offinterlineskip
\hbox{\boxtext{\phantom{ X}}\boxtext{\phantom{ X}}\boxtext{\phantom{ X}}\boxtext{\phantom{ X}}}
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}

\item Coordinates of the boxes

\bigskip
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\centerline{
\vbox{\offinterlineskip
\hbox{\boxtext{{\normalsize(1,1)}}\boxtext{{\normalsize(1,2)}}\boxtext{{\normalsize(1,3)}}\boxtext{{\normalsize(1,4)}}}
\hbox{\boxtext{{\normalsize(2,1)}}\boxtext{{\normalsize(2,2)}}}
\hbox{\boxtext{{\normalsize(3,1)}}}
}
}


\item The expression ``for all $(i,j)\in \lambda$'' means:

\bigskip

``for all pairs $(i,j)$ such that $(i,j)$ are coordinates of boxes in the Young diagram of $\lambda$''


% ---------------------------------------------------------------

\slide{Schur functions. Part Three: Transposed Partitions}

\item Definition: If $\lambda=(4,2,1)$, $\lambda'=(3,2,1,1)$

\bigskip
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% -----------------------------------------------------------
\slide{Schur functions. Part Four: Young tableaux}


\item Definition: (Semistandard) Young tableaux (SSYT)

\bigskip

A Young diagram filled with numbers, strictly increasing by columns,
nonstrictly by rows

\item Example: Young tableaux: Two that are and one that is not!

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\item Definition: $x^T$, where $T$ is a SSYT

$$
x_1^2 x_2^3 x_3 x_4\hskip2truecm
x_1x_2x_3x_4x_5x_6x_7\hskip1.5truecm\phantom{X}
$$






%-----------------------------------------------------------

\slide{Schur functions. Part Five: Definition}

$$
s_{\lambda}(x_1,\ldots,x_m)\equiv\sum_T x^T
$$

where the summation is over all Young tableaux of shape $\lambda$ filled
with the numbers $1,2,\ldots,m$

\bigskip

Default case: For $\lambda=(0), s_\lambda\equiv 1$.

\item Example 1: $\lambda=(1)$

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$\cdots$
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\boxtext{m}
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$$
s_{(1)}(x_1,\ldots,x_m)= x_1+\cdots+x_m
$$


\item Example 2: $\lambda=(2)$

\bigskip
\centerline{
\boxtext{$i$}\boxtext{$j$}
}


$$
s_{(2)}(x_1,\ldots,x_m)= \sum_{i\le j} x_ix_j
$$


% ------------------------------------------------------------------
\slide{Schur functions. Part Five: Definition}


\item Example 3: $\lambda=(1,1)$

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$$
s_{(1,1)}(x_1,\ldots,x_m)= \sum_{i< j} x_ix_j
$$



\item Example 4: Schur function, $m=3, \lambda=(2,1)$


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$$
s_{(2,1)}(x_1,x_2,x_3)=2x_1x_2x_3+x_1^2x_2+x_1x_2^2+
x_1^2x_3+x_1x_3^2+x_2^2x_3+x_2x_3^2
$$

% -----------------------------------------------------------
\slide{Schur functions. Part Six: Recursive formula}


\item Key in computing the Schur/Zonal polynomials!

\item Observe that $m$ can only occur in a ``rightmost square.''


\bigskip
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The rightmost squares form a ``horizontal strip.''

\item Therefore

$$s_\lambda(x_1,\ldots,x_m)=\sum_{\mu} s_\mu(x_1,\ldots,x_{m-1})
x_m^{|\lambda|-|\mu|}
$$

where the summation is over all partitions $\mu$ which ``fit inside''
$\lambda$ and such that $\lambda-\mu$ is a horizontal strip.

\item Example:
$$
s_{(2)}(x_1,x_2) = s_{(2)}(x_1) + s_{(1)}(x_1) x_2 + s_{(0)}(x_1)x_2^2 =
x_1^2 + x_1x_2 + x_2^2
$$

% ------------------------------------------------------------------

\slide{Zonal and Jack polynomials}

\item That $\beta$ parameter again!

\item In combinatorics people use $\alpha=2/\beta$ instead

\item Schur $\beta=2, \alpha=1$

\item Zonal $\beta=1, \alpha=2$

\item Jack---general $\beta, \alpha$


% ----------------------------------------------------------

\slide{Hook lengths}

\item Definition: For $(i,j)\in \lambda$
$$
h_\lambda(i,j)\equiv
\lambda_j'-i+\lambda_i-j+1
$$

\item Example: For $\lambda=(4,2,1)$, $h_\lambda(1,2)=4$

\bigskip
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\item In general the hook lengths are


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\hbox{\boxtext{ 3}\boxtext{ 1}}
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\item $\alpha$--modifications: Upper and lower hook lengths
\begin{eqnarray*}
h_\lambda^{*}(i,j)&\equiv&\lambda_j'-i+\alpha(\lambda_i-j+1)\\
h_{*}^\lambda(i,j)&\equiv&\lambda_j'-i+1+\alpha(\lambda_i-j)
\end{eqnarray*}



% -------------------------------------------------------------

\slide{Jack function}

\item Definition: Jack function (for any $\alpha$; plug in $\alpha=2$ for
the Zonals)

$$
J_\kappa(x_1,x_2,\ldots,x_m)=\sum_\mu
J_\mu(x_1,x_2,\ldots,x_{m-1})
x_m^{|\kappa/\mu|}\xi_{\kappa\mu},
$$
where the summation is over all $\mu\le\kappa$ such that
$\kappa/\mu$ is a horizontal strip, and
\begin{equation}
\xi_{\kappa\mu}\equiv\frac{
 \prod_{(i,j)\in \kappa} B_{\kappa\mu}^\kappa(i,j)
}{
\prod_{(i,j)\in \mu} B_{\kappa\mu}^\mu(i,j)
},
\mbox{\hskip0.2in where\hskip0.2in}
B_{\kappa\mu}^\nu(i,j)\equiv\left\{
\begin{array}{ll}
h_\nu^{*}(i,j), & \mbox{if }\kappa_j'=\mu_j'; \\
h_{*}^\nu(i,j), & \mbox{otherwise,}
\end{array}
\right.
\nonumber
\end{equation}

($h_\kappa^{*}$ and $h_{*}^\kappa$ are the upper and lower hook lengths)

\item $\xi_{\kappa\mu}=$ messy-to-describe, yet easy-to-compute
rational function of $\alpha$

\bigskip
\bigskip

\item Caveat: Need to be careful about normalization---see our paper with 
Edelman

% --------------------------------------------------------------

\slide{References}

\item R.~Stanley, Some combinatorial properties of Jack symmetric
functions, Adv. Math. 77 (1989), no. 1, 76--115.

\item I.~G.~Macdonald, Symmetric functions and Hall polynomials, Second
ed.,
Oxford University
Press, New York, 1995.

\item P.~Koev, A.~Edelman, The Efficient Evaluation of the Hypergeometric
Function of a Matrix
Argument, Math. Comp. 75 (2006), 833-846.

\item Ioana Dumitriu's MOPS package

\item My website: Software for computing (symbolically) Schur, Jack,
Zonals, Hypergeometric functions of matrix argument

\begin{center}
\verb+http://math.mit.edu/~plamen+
\end{center}


\end{itemize}

}

\end{document}

