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\def\({\left(}
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\topmatter
\title
      Hypergeometric Functions
         Associated with Positive Roots
\endtitle
\rightheadtext{Hypergeometric functions associated with positive roots}
\leftheadtext{I.~M.~Gelfand, M.~I.~Graev, and A.~Postnikov}
\author  {Israel M. Gelfand \medskip
{\rm Department of Mathematics\\
Rutgers University\\
New Brunswick, NJ 08903, U.S.A.\linebreak
E-mail: igelfand\@math.rutgers.edu}
\bigskip 
 Mark I. Graev\medskip 
{\rm Department of Mathematics\\
Research Institute for System Studies RAS\\
   23 Avtozavodskaya St, Moscow 109280, Russia\\
 E-mail: Graev\@systud.msk.su}
\bigskip
 Alexander Postnikov\medskip 
{\rm Department of Mathematics\\
Massachusetts Institute of Technology\\
Cambridge, MA 02139, U.S.A.\linebreak
E-mail: apost\@math.mit.edu}}
\endauthor
\date January  18, 1995 \enddate
%\dedicatory \enddedicatory
%\thanks \endthanks
%\keywords \endkeywords
%\subjclass \endsubjclass

\abstract
In this paper we study the hypergeometric system on  unipotent matrices. 
This system gives a holonomic $D$-module. We find the number of independent
solutions of this system at a generic point.
 This number  is equal to 
the famous Catalan number.
An explicit basis  of $\Gamma$-series in solution space of this system
is constructed in the paper. 
We also consider restriction of this system to certain strata. 
We introduce several combinatorial constructions with trees, polyhedra,
and triangulations related to this subject.	
\endabstract


\toc\widestnumber\head{10}
%\specialhead{} Introduction \endspecialhead
\head 1. General Hypergeometric Systems \endhead
\head 2. Hypergeometric Systems on  Unipotent Matrices \endhead
\head 3. Integral  Expression for Hypergeometric Functions   
 \endhead
\head 4. $\Gamma$-series and Admissible Bases  \endhead
\head 5. Admissible Trees \endhead
\head 6. Standard Triangulation of $P_n$   \endhead
\head 7. Coordinate Strata \endhead
\head 8.  Face Strata  \endhead
\head 9. Standard Triangulation of $P_{IJ}$ \endhead
\head 10. Examples  \endhead
\head 11. Concluding Remarks and Open Problems \endhead
\endtoc


\endtopmatter


\document
%\specialhead{} Introduction \endspecialhead






\head 1. General Hypergeometric Systems \endhead


In this paper we use the following notation:
 $ [a,b]:=\{a,a{+}1,\dots,b\}$ and $[n]:=[1,n]$.


Recall several definitions and facts from the theory of general hypergeometric functions (see \cite{GGZ, GZK, GGR2}).

Consider the following action 
 of the complex $n$-dimensional torus 
$T=({\Bbb C}^\ast)^n$ with coordinates $t=(t_1,t_2,\dots,t_n)$
on the space ${\Bbb C}^N$ 
 $$ 
     x=(x_1,x_2,\dots,x_N)\longmapsto x\cdot t=(x_1t^{a_1},\dots ,x_Nt^{a_N}),
\tag 1.1 $$
where $a_j=(a_{1j},\dots,a_{nj})\in{\Bbb Z}^n,$ $ j=1,2,\dots,N$
and $t^{a_j}$ denotes $t_1^{a_{1j}}\dots t_n^{a_{nj}}$.

\definition{Definition 1.1} The {\it General Hypergeometric  System\/}
associated with action of torus (1.1) is the following system of 
differential equations on ${\Bbb C}^N$
$$\align
       \sum_{j=1}^N a_{ij}x_j \frac{\partial f}{\partial x_j}&=\alpha_i f,
\qquad i=1,2,\dots,n; \tag 1.2 \\
     \prod_{j:\,l_j>0}\(\frac\partial{\partial x_j}\)^{l_j}f&=
        \prod_{j:\,l_j<0}\(\frac\partial{\partial x_j}\)^{-l_j}f, \tag 1.3
\endalign
$$
where $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)\in{\Bbb C}^n$ and
$l=(l_1,l_2,\dots,l_N)$ ranges over the lattice $L$ of integer
 vectors such that
$l_1a_1+l_2a_2+\dots+l_Na_N=0$.

Solutions of the system (1.2), (1.3) are called {\it hypergeometric
 functions\/}
on ${\Bbb C}^N$ associated with action of torus (1.1).
The numbers $\alpha_i$ are called
 {\it exponents.}
\enddefinition

\remark{Remark 1.2} Equations (1.2) are equivalent to the following 
 homogeneous conditions
$$ f(x\cdot t)= t^\alpha f(x),\tag 1.4
$$
where $t=(t_1,t_2,\dots,t_n)\in T$ and $t^\alpha=
{t_1}^{\alpha_1}{t_2}^{\alpha_2}\dots {t_n}^{\alpha_n}$.
\endremark

\remark{Remark 1.3} For generic $\alpha$ system (1.2), (1.3) is equivalent
to the subsystem, where $L$ ranges over any set of generators for 
the lattice $L$.
\endremark

By $A$ denote the set of integer vectors $a_1,a_2,\dots,a_N$. Let $H_A$
be the sublattice in $\Bbb Z^n$ generated by $a_1,a_2,\dots,a_N$ and 
$m=\Dim H_A$ be the dimension of $H_A$.
 Let  $P_A$ denote the convex
hull of the origin 0 and $a_1,a_2,\dots,a_N$. Then $P_A$ is a polyhedron with 
vertices in the lattice $H_A$.

Let $\text{Vol}_{H_A}$ be the form of volume on the space 
$H_A\otimes_{\Bbb Z}{\Bbb R}$ such that  volume of the identity cube
is equal to 1. The volume of a polyhedron with vertices in the lattice $H_A$
times $m!$ is an integer number. In particular, 
$m! \Vol_{H_A}P_A$
 is integer.

\proclaim{Theorem 1.4} The general hypergeometric system (1.2), (1.3)
gives a holonomic $D$-module. The number of linearly independent solutions of
this system in 
a neighborhood of a generic point
is equal to  $m! \Vol_{H_A}P_A$.
\endproclaim

If there exist an integer covector $h$ such that
$$ h(a_j)=1\qquad \text{for all}\qquad j=1,2,\dots,N
\tag 1.5 $$
then we call the corresponding system (1.2), (1.3)
{\it flat\/} or {\it nonconfluent.}

Theorem~1.4 in nonconfluent case was proved in \cite{GZK}.
Very close results were found by Adolphson in \cite{Ad},
but his technique is quite different from ours.

In this paper we study one special case of systems (1.2), (1.3)
when condition (1.5) does not hold. We define these systems
in the following section.
\bigskip





\head 2. Hypergeometric System on  Unipotent
Matrices  \endhead










Let $R\subset \Bbb Z^n$ be a {\it root system\/} and $R^{+}\subset R$ 
be the set of {\it positive roots} (see \cite{Bo}).
Then we can define the hypergeometric system (1.2), (1.3) associated
with the set of integer vectors $A=R^+$.

We consider the case of the root system $A_n$ in more details.

Let $\epsilon_0,\epsilon_1,\dots,\epsilon_n$ be the standard basis in
the lattice $\Bbb Z^{n{+}1}$. The root system $A_n$ is the set
of all vectors (roots) $e_{ij}=\epsilon_i-\epsilon_j$, $i\ne j$.
Let $A=A_n^+$ be the set of all positive roots 
$A=\{e_{ij}\in A_n:0{\le} i{<}j{\le n}\}$.

It is clear that 
positive roots generate the $n$-dimensional lattice $H_A\simeq\Bbb Z^n$ of
 all vectors $v=v_0\epsilon_0+v_1\epsilon_1+\dots+v_n\epsilon_n$,
$ v_i\in\Bbb Z$ such that $v_0+v_1+\dots+v_n=0$.


By $Z_n$ denote the group of unipotent matrices of
order $n{+}1$, i.e. the group of upper triangular matrices $z=(z_{ij})$,
$0{\le} i{\le} j{\le} n$
with 1's on the
diagonal $z_{ii}=1$. 





The $n$-dimensional torus $T$ presented as the group
of diagonal matrices $t=\diag(t_0,t_1,\dots,t_n)$, 
$ t_0\cdot t_1\dots t_n=1$
acts on $Z_n$ by conjugation $z\in Z_n\to tzt^{-1}$,
or in coordinates
$$ z=\{z_{ij}\}\longmapsto \{z_{ij}t_it_j^{{-}1}\}. \tag 2.1
$$


Clearly, action of torus (1.1) associated with the set of
vectors $A=A_n^+$ is the same as  action  (2.1).
Here $N=\binom {n{+}1}2$ and $z_{ij}$, $ 0{\le} i{<}j{\le} n$ are
coordinates in $\Bbb C^N$.













The main object of this paper is the hypergeometric system
 associated with action (2.1). Write down this system 
explicitly.

\definition{Definition 2.1} The {\it  Hypergeometric System
on the Group of Unipotent Matrices\/}
 is the following system of differential equation 
on the space  $Z_n \simeq \Bbb C^N$  with coordinates 
$z_{ij}$, $ 0{\le} i{<}j{\le} n$
$$
\align
-\sum_{i=0}^{j{-}1}z_{ij}\,\frac{\partial f}{\pa z_{ij}}
&+\sum_{k=j{+}1}^n z_{jk}\frac{\pa f}{\pa z_{ij}}=\alpha_j f,
\qquad j=0,1,\dots,n;
\tag 2.2 \\
 \frac{\pa f}{\pa z_{ik}}&=\frac{\pa^2 f}{\pa z_{ij}\,\pa z_{jk}},
\qquad 0{\le} i{<}j{<}k{\le} n, \tag 2.3	
\endalign
$$
where $\alpha=(\alpha_0,\alpha_1,\dots,\alpha_n)\in
\Bbb C^{n{+}1}$ is a vector such that $\sum\alpha_j=0$.

Solutions of system (2.2), (2.3) are called {\it hypergeometric functions
on the group of unipotent matrices.}
\enddefinition

In order to prove that system (2.2), (2.3) is a special
 case of the system (1.2), (1.3) we need the following simple 
lemma.

\proclaim{Lemma 2.2} It follows from equations (2.3) that
$$
  \prod_{(i,j):\,l_{ij}>0}\(\frac\partial
{\partial z_{ij}}\)^{l_{ij}}f=
       \prod_{(i,j):\,l_{ij}<0}\(\frac\partial
{\partial z_{ij}}\)^{-l_{ij}}f,  
\tag 2.4 $$
for all $l=(l_{ij})$, $ 0{\le} i{<}j{\le} n$, $l_{ij}\in\Bbb Z$
such that $\sum_i l_{ij}-\sum_k l_{jk}=0$, $ j=0,1,\dots,n$.
\endproclaim

\demo{Proof}
It follows from (2.3) that
$$
\frac{\pa f}{\pa z_{ij}}=
\frac{\pa^{j-i}f}
{\pa z_{ii{+}1}\,\pa z_{i{+}1i{+}2}\dots\pa z_{j{-}1j}}
$$
Now change in (2.4) all occurrences of $\frac\pa{\pa z_{ij}}$
to $\frac{\pa^{j-i}}
{\pa z_{ii{+}1}\,\pa z_{i{+}1i{+}2}\dots\pa z_{j{-}1j}}$.
We get the same expressions in LHS and in RHS.
\enddemo

Let $P_n=P_{A_n^+}$ be the convex hull of the origin 0
and of $e_{ij}$, $ 0{\le} i{<}j{\le} n$. The first part of
the following theorem is a special
 case of Theorem~1.4.

\proclaim{Theorem 2.3} 
\roster
\item The hypergeometric system (2.2), (2.3) gives a holonomic
$D$-module.
The number of linearly independent solutions of this system in a neighborhood
of a generic point is equal to $n! \Vol P_n$.
\item $n! \Vol P_n$ is equal to the Catalan number
$$
C_n= \frac1{n+1}\binom{2n}n.
$$
\endroster
\endproclaim
\medskip





\head 3. Integral Expression for Hypergeometric Functions  \endhead

In this section we present an integral expression for hypergeometric functions on
unipotent matrices (see \cite{GG1}).


 Consider the following integral
$$
f(z)=\int_C\exp\(\sum z_{ij}t_it_j^{-1}\)\,t^{-\alpha}\frac{dt}{t},\tag 3.1
$$
where the sum in exponent is over $0{\le} i{<}j{\le} n$;
$t$
is a point of
torus $T=\{(t_0,\dots,t_n):t_0\cdot\dots\cdot t_n=1\}\simeq(\Bbb C^*)^n$;
$t^{-\alpha}\,dt/t=t_1^{-\alpha_1}\dots t_n^{-\alpha_n}\,dt_1/t_1\dots dt_n/t_n$;
and
$C$ is a real $n$-dimensional cycle in $2n$-dimensional space
$T$.

\proclaim{Theorem 3.1} The function $f(z)$ given by integral (3.1)
 is a solution of the hypergeometric system (2.2), (2.3).
\endproclaim
\medskip












\head 4. $\Gamma$-series and Admissible Bases \endhead


In this section we construct an explicit basis in the
solution space of system (1.2), (1.3). In case of nonconfluent
systems this construction was given in \cite{GZK}.
In this section we basically follow \cite{GZK}.

Recall that $A=\{a_1,a_2,\dots,a_N\}$, where $a_j\in \Bbb Z^n$. 
Without loss of
generality we can assume that vectors $a_j$ generate the lattice $\Bbb Z^n$,
i.e. $ H_A=\Bbb Z^n$.


Let $\gamma=(\gamma_1,\gamma_2,\dots,\gamma_N)\in \Bbb C^N$.
Consider the following formal series
$$
\Phi_\gamma(x)=\sum_{l\in L}\frac{x^{\gamma+l}}{\prod_{j=1}^N 
\Gamma(\gamma_j+l_j+1)},  \tag 4.1
$$
where $x=(x_1,x_2,\dots,x_N)$,  
$L$ is the lattice such as in Definition 1.1, and $x^{\gamma+l}=
\prod_{j=1}^N x_j^{\gamma_j+l_j}$.

\proclaim{Lemma 4.1} The series $\Phi_\gamma(x)$ formally satisfies
 system (1.2), (1.3) with $\alpha=\sum_j \gamma_j a_j$.
\endproclaim

For a fixed vector of exponents $\alpha=(\alpha_1,\dots,\alpha_n)$ 
 the vector $\gamma=(\gamma_1,\dots,\gamma_N)$ ranges over
the affine $(N-n)$-dimensional plane 
$\Pi(\alpha)=\{(\gamma_1,\dots,\gamma_N):
\sum_j\gamma_j a_j=\alpha\}$. In this section we
 construct several vectors $\gamma$ such
that all series $\Phi_\gamma(x)$ converge in certain  neighborhood and
form a basis in the space of solutions of system (1.2), (1.3) in this
neighborhood. 

A subset $\I\in [N]$ is called a {\it base\/}
if vectors $a_j,$ $j\in \I$ form a basis of 
the linear space $H_A\otimes\Bbb R$. So we get a {\it matroid\/} on the set $[N]$.
Let $\Delta_\I$ be the $n$-dimensional simplex with vertices 0 and $a_j$,
$j\in \I$.

Let $\I$ be a base.  
By $\Pi(\alpha,\I)$ denote the set of $\gamma\in\Pi(\alpha)$
such that $\gamma_j\in\Bbb Z$ for $j\notin \I$. It is clear that for every
$l\in L$ (see Definition 1.1) $\Phi_\gamma(x)=\Phi_{\gamma+l}$.

The following lemma was proven in \cite{GZK}.

\proclaim{Lemma 4.2} Let $\I$ be a base. Then
 $ |\Pi(\alpha,\I)/L|=n!\Vol(\Delta_\I)$.
\endproclaim 	






\definition{Definition 4.3} We call a base $\I\in [N]$ {\it admissible\/}
if the $(n-1)$-dimensional simplex with vertices $a_j$, $j\in \I$
belongs to the boundary $\pa P_A$ of the polyhedron $P_A$.
In this case the  simplex $\Delta_\I   $ is also called {\it admissible\/}.
\enddefinition 

\remark{Remark 4.4} If vectors $a_j$ satisfy condition (1.5)
then all bases are admissible.
\endremark

Let $B=\{b_1,b_2,\dots,b_{N-n}\}$ be a $\Bbb Z$-basis in the lattice $L$.
We say that a base $\I$ is 
{\it compatible\/} with a basis $B$  if
whenever $l=(l_1,\dots ,l_N)\in L$ such that $l_j\ge 0$ for $j\notin \I$
then $l$ can be expressed as $l=\sum \lambda_k b_k$, 
where all $\lambda_k\ge 0$.
Clearly, the set $\Pi_B(\alpha,\I)=\{\gamma\in\Pi(\alpha,\I): 
\gamma=\sum \lambda_k b_k, \text{ where } 0{\le} \lambda_k{<}1\}$ is a set
of representatives  in  $\Pi(\alpha,\I)/L$.

Let $y_k=x^{b_k}$, $k=1,2\dots,N-n$. 

\proclaim{Proposition 4.5} Let an admissible base $\I$ be compatible with
a basis $B$. Then for all $\gamma\in\Pi_B(\alpha,\I)$ the series
$\Phi_\gamma(x)$ is of the form
$\Phi_\gamma(x)=x^\gamma \sum_m c(m) y^m$, where the sum is over
$m=(m_1,\dots,m_{N-n})$, $m_k\ge 0$. 
  The series $\sum  c(m) y^m$ converges for sufficiently small
$|y_k|$.
\endproclaim

\demo{Proof}
Let $b_k=(b_{k1},\dots,b_{kN})\in L$, $k=1,\dots,N{-}n$.
By definition,
$\Phi_\gamma(x)=x^\gamma \sum_m c(m) y^m$, where
$c(m)=\prod_j \Gamma(\gamma_j+\sum_k m_k b_{kj}+1)^{-1}$,
$m=(m_1,\dots,m_{N-n})\in \Bbb Z^{N{-}n}$.
Let $\gamma\in \Pi_B(\alpha,\I)$. Then $\gamma_j+\sum_k m_k b_{kj}+1\in\Bbb Z$,
for $j\not\in \I$.
Hence, if $c(m)\ne 0$ then  
$\gamma_j+\sum_k m_k b_{kj}+1\ge 0$,  $j\not\in \I$.
Since $\I$ is compatible with $B$, we can deduce that $c(m)\ne 0$ only if
$m_k\ge 0$, $k=1,\dots, N{-}n$ (see details in \cite{GZK}).
Convergence of the series $\sum c(m) y^m$ follows from the next lemma.
\enddemo


\proclaim{Lemma 4.6} Let
$c(m)=\prod_j \Gamma(\mu_j(m)+\gamma_j+1)^{-1}$, $m=(m_1,\dots,m_r)$,
$m_k\ge 0$, where $\mu_j$ are linear functions of $m$ such that
$\sum\mu_j(m)=s_1m_1+\dots +s_r m_r$, $s_k\ge 0$.
Then $|c(m)|\le R\,c_1^{m_1}\dots c_r^{m_r}$
for some positive constants $R,c_1,\dots,c_r$.
\endproclaim

It is not difficult to prove this Lemma using Stiltjes formula.



Thus, by Proposition 4.5 for  every admissible base
$I$ we have  $n!\Vol(\Delta_\I)$
series $\Phi_\gamma(x)$, $\gamma\in\Pi_B(\alpha,\I)$
 with nonempty common  convergence domain.


\remark{Remark 4.7} It can be shown that if $\gamma\in \Pi(\alpha,\I)$,
where $\I$ is not admissible, then $\Phi_\gamma (x)$ diverges.
\endremark


Recall that $P_A$ is the convex hull of 0 and $a_j$, $j=1,2,\dots,N$.  

\definition{Definition 4.8} The set of bases $\T$ is called a {\it local
triangulation\/} of $P_A$ if 
\roster
\item $\cup_{\I\in\T}\Delta_\I=P_A$;
\item  $\Delta_{\I_1}\cap\Delta_{\I_2}$ is
the common face of $\Delta_{\I_1}$ and $\Delta_{\I_2}$ 
for all $\I_1,\I_2\in\T$.
\endroster
\enddefinition

We call such triangulation $\T$ local because all simplices $\Delta_{\I}$,
  $\I\in\T$
contain the origin~0.

\remark{Remark 4.9} Note that if  $\T$ is a local triangulation then
all bases $\I\in \T$  are admissible
\endremark

\definition{Definition 4.10} A local triangulation  $\T$ is called
{\it coherent\/} if there exist a piecewise linear function $\phi$
on
$P_A$ such that  $\phi$ is linear on simplices $\Delta_\I$, $\I\in \T$
and  $\phi$ is strictly convex on $P_A$.
\enddefinition


\proclaim{Lemma 4.11}
There exists a coherent local triangulation
of $P_A$.
\endproclaim

\proclaim{Lemma 4.12}
Let $\T$ be a coherent local triangulation of $P_A$.
Then there exist a basis $B$ of $H_A$ such that $B$ is compatible with
every base $\I$ in $\T$.
\endproclaim

\proclaim{Theorem 4.13} Let $\T$ be a coherent local triangulation of 
$P_A$; and $B=\{b_1,b_2,\dots\mathbreak\dots,b_{N-n}\}$ a basis such as in 
Lemma 4.12.
Let $y_k=x^{b_k}$. Then for every $\gamma\in\Pi_B(\alpha,\I)$, $\I\in\T$
the series $\Phi_\gamma(x)$ is equal $x^\gamma$ times a series of
variables $y_k$, which converges for sufficiently small $|y_k|$.
If exponents $\alpha_1,\alpha_2,\dots, \alpha_n$ are generic
then all these series $\Phi_\gamma(x)$ are linearly independent.
\endproclaim

Hence, for generic $\alpha=(\alpha_1,\alpha_2,\dots, \alpha_n)$ we constructed
$n!\Vol(P_A)$ independent solutions of system (1.2), (1.3),
which converge in common domain. Therefore, by Theorem~1.4,
these series form a basis in the space of solutions of system (1.2), (1.3).
\bigskip








\head 5. Admissible Trees \endhead






In this section we describe admissible bases in
 the case of the hypergeometric system (2.2), (2.3). 

It is well known
that a subset $\I\subset
\{(i,j):0{\le} i{<}j{\le} n\} $
 is a base in the set of positive roots $A=A_n^+$
if and only if $\I$  is the set of edges of
a tree $T_\I$ on $[0,n]$. 



\definition{Definition 5.1} A tree $T$ on the set $[0,n]$ is called
{\it admissible\/} if there are no $0{\le} i{<}j{<}k{\le} n$ 
such that both $(i,j)$
and $(j,k)$ are edges of $T$.
\enddefinition

\proclaim{Proposition 5.2} A subset  $\I\subset
\{(i,j):0{\le} i{<}j{\le} n\} $ is an admissible base in $A=A_n^+$
if and only if $T_\I$ is an admissible tree.
\endproclaim

\proclaim{Lemma 5.3}  $n!\Vol \Delta_\I=1$
for any base $\I$.
\endproclaim

Therefore, by Lemma 4.2 $|\Pi(\alpha,\I)/L|=1$ and
by Proposition~4.5 for every admissible tree $T$ 
we have a series $\Phi_T (z)=\Phi_\gamma(z)$,
where $\gamma\in\Pi(\alpha,\I)$, $T=T_\I$.
 The series $\Phi_T (z)$ converges in some domain and
 presents a solution of the system (2.2), (2.3).


There exists a  formula for the number of all admissible
trees on the set $[0,n]$. 

\proclaim{Theorem 5.4} The number $F_n$ of admissible trees on the set
of vertices $[0,n]$ is equal to
$$
F_n=\frac{1}{2^n (n{+}1)} \sum_{k=1}^{n{+}1}\binom {n{+}1}k k^{n}.
$$
\endproclaim

The proof of this formula is given in \cite{Po}.
 
First few numbers $F_n$ are given below.
\bigskip
\hrule 
\line{\vrule\hfill\vbox{\hbox{\strut $n$}\hbox{$F_n\,$\strut}}\hfill\vrule%
\hfill
\vbox{\hbox{\strut 0}\hbox{\strut 1}}\hfill\vrule\hfill
\vbox{\hbox{\strut 1}\hbox{\strut 1}}\hfill\vrule\hfill
\vbox{\hbox{\strut 2}\hbox{\strut 2}}\hfill\vrule\hfill
\vbox{\hbox{\strut 3}\hbox{\strut 7}}\hfill\vrule\hfill
\vbox{\hbox{\strut 4}\hbox{\strut 36}}\hfill\vrule\hfill
\vbox{\hbox{\strut 5}\hbox{\strut 246}}\hfill\vrule\hfill
\vbox{\hbox{\strut 6}\hbox{\strut 2104}}\hfill\vrule\hfill
\vbox{\hbox{\strut 7}\hbox{\strut 21652}}\hfill\vrule}
\hrule
%\medskip
%\hrule
%\line{\vrule\hfill\vbox{\hbox{\strut $n$}\hbox{$F_n\,$\strut}}\hfill\vrule%
%\hfill
%\vbox{\hbox{\strut 8}\hbox{\strut 260720}}\hfill\vrule\hfill
%\vbox{\hbox{\strut 9}\hbox{\strut 3598120}}\hfill\vrule\hfill
%\vbox{\hbox{\strut 10}\hbox{\strut 56010096}}\hfill\vrule\hfill
%\vbox{\hbox{\strut 11}\hbox{\strut 971055240}}\hfill\vrule}
%\hrule
\bigskip\medskip





\head 6. Standard Triangulation of $P_n$   \endhead

Recall that $P_n$ is the convex hull of 0
and $e_{ij}$, $0{\le} i{<}j{\le} n$.
 
In this section we construct a coherent triangulation of the polyhedron $P_n$.
 This will give us an explicit basis in the solution space of
system (2.2), (2.3).

Let $T$ be a tree on the set $[0,n]$. We say that two edges $(i,j)$
and $(k,l)$ in $T$ form an {\it intersection\/} if $i{<}k{<}j{<}l$.

\definition{Definition 6.1} A tree $T$ on the set $[0,n]$ is
called {\it standard\/} if $T$ is admissible and does not have
intersections. The corresponding base $\I\subset\{(i,j):0{\le} i{<}j{\le} n\}$
is also called {\it standard\/}.
\enddefinition




\example{Example 6.2} All standard trees for $n=0,1,2,3$ are shown on
Figure 6.1.
\midinsert
\vskip 10pt
\line{\hfil \epsfysize=6.0cm \epsfbox{figure61.eps}\hfil }
\vskip 10pt
\botcaption{Figure 6.1} Standard trees.
\endcaption
\endinsert
\endexample 

\proclaim{Theorem 6.3} The set $\T_n$ of standard bases forms a coherent
local triangulation of the polyhedron $P_n$.
\endproclaim

\proclaim{Theorem 6.4} The number of standard trees on the set $[0,n]$
is equal to the Catalan number 
$$C_n=\frac1{n+1}\binom{2n}n.$$
\endproclaim

As a consequence of these two theorems we get Theorem~2.3.(2).

\demo{Proof of Theorem 6.4}
Construct by induction an explicit 1--1 correspondence $\psi_n$ between the set
$\text{ST}_n$ of standard trees on $[0,n]$ and the set
 $\text{BT}_n$ of binary trees
with $n$ unmarked vertices
$\psi_n:\text{ST}_n\to\text{BT}_n.$

If $n=1$ then $\psi_1$ maps a unique element of $\text{ST}_1$ to a unique 
element of $\text{BT}_1$.

Let $n>1$. Every standard tree $T\in\text{ST}_n$ has the edge $(0,n)$.
Delete this edge. Then $T$ splits into two standard trees $T_1\in\text{BT}_k $
 and $T_2\in\text{BT}_l$, $k+l+1=n$ on the sets $[0,k]$ and $[k{+}1,n]$.
Let as define $\psi_n(T)$ as the binary tree whose  left and  right
 branches are equal to
$\psi_k(T_1)$ and $\psi_l(T_2)$ correspondingly. 
See example on Figure 6.2.

It is  well known  that the number of binary trees is equal to the
Catalan number (e.g. see \cite{SW}).
\enddemo

\midinsert
\vskip 10pt
\line{\hfil \epsfysize=1.5cm \epsfbox{figure62.eps}\hfil }
\vskip 10pt
\botcaption{Figure 6.2} Bijection between standard and binary trees.
\endcaption
\endinsert


Now prove Theorem~6.3.

\demo{Proof of Theorem 6.3}
Recall that $\epsilon_0,\epsilon_1,\dots,\epsilon_n$ is the standard basis
in $\Bbb Z^{n{+}1}$; and $e_{ij}=\epsilon_i-\epsilon_j$. 


Let $\widetilde{P}_n\subset\Bbb Z^{n{+}1}\otimes \Bbb R$
 denote the cone with vertex at 0 generated 
by all positive roots $e_{ij}$, $i<j$.
Let $\widetilde{\Delta}_\I$ denote the simplicial cone generated by
$e_{ij}$, $(i,j)\in \I$, where $\I$ is a base 
(the  cone over the simplex
 $\Delta_\I$).
 
First, prove that the collection of cones $\widetilde{\Delta}_\I$, where 
$\I$ range over all standard bases, is a conic triangulation of
$\widetilde{P}_n$. Then it  follows that $\T_n$ is a local triangulation.

It is not difficult to show that the cone $\widetilde{P}_n$ is the set 
of $v=(v_0,v_1,\dots,v_n)\in\Bbb R^{n{+}1}$
such that
$$\align
v_0+v_1+\dots+v_i&\ge 0,\qquad i=1,2,\dots,n-1; \tag 6.1   \\
v_0+v_1+\dots+v_n&=0.   \tag 6.2
   \endalign
$$

We must show that  every generic point $v$ subject to (6.1), (6.2)
can be uniquely
 presented in the form
$$v=\sum_{(ij)\in \I}\rho_{ij}e_{ij},\quad \rho_{ij}\ge 0,  \tag 6.3
$$
for some standard base $\I$.

Prove it by induction on $n$.

Let $v'=(v_0',v_1',\dots,v_{n{-}1}')\in\Bbb R^n$ be a vector such that
$v_i'=v_i$, $i=0,1,\dots,n{-}2$, and $v_{n{-}1}'=v_{n{-}1}+v_n$.
Then $v'\in\widetilde{P}_{n{-}1}$. By induction we may assume
that $v'$ is expressed in the form 
$$v'=\sum_{(ij)\in \I'}\rho_{ij}'e_{ij},\quad \rho_{ij}'\ge 0, 
$$
 for a standard base $\I'\subset
\{(i,j):0{\le} i{<}j{\le} n{-}1\}$.

Let $i_1{<}i_2{<}\dots{<}i_s$ be all 
vertices of $T'=T_{\I'}$ connected with the
vertex $n{-}1$ in $T'$.

Consider two cases.

1. $v_{n{-}1}\ge 0$. Define $\I=\I'\cup \{(n{-}1,n)\}\cup
\{(i_k,n):k\in[s]\}\setminus \{(i_k,n{-}1):k\in[s]\}$. And
$\pho_{ij}=\pho_{ij}'$ for $0{\le} i{<}j{\le} n{-}2$; 
$\pho_{i_kn}=\pho_{i_kn{-}1}'$
for $k\in[s]$;  $\pho_{n{-}1n}=v_{n-1}$.
Then we get  expression (6.3) for $v$.

2. $v_{n{-}1}<0$.  Then $-v_n\le\sum_{k=1}^s \pho_{i_kn{-}1}'$. Let $t$ be
 the minimal integer  $0{\le} t{\le} s$ such that 
$\sum_{k=1}^t \pho_{i_kn{-}1}'\ge- v_n$.
Then define $\I=\I'\cup\{(i_k,n):k\in[t]\}\setminus
\{(i_k,n{-}1):k\in[t{-}1]\}$.
 And $\pho_{ij}=\pho_{ij}$ for
$0{\le} i{<}j{\le} n{-}2$; $\pho_{i_kn}=\pho_{i_kn{-}1}'$ for $k\in[t{-}1]$;
$\pho_{i_t n}=-\sum_{k=1}^{t{-}1} \pho_{i_kn{-}1}'-v_n$;
$\pho_{i_kn{-}1}=\pho_{i_kn{-}1}'$ for $k\in[t{+}1,s]$;
 $\pho_{i_tn{-}1}=-\sum_{k=t{+}1}^s \pho_{i_kn{-}1}'-v_{n{-}1}$.
Then we get  expression (6.3) for $v$.  
 

Therefore, $\T_n$ is a local triangulation.

Prove that
$\T_n$ is coherent triangulation (see Definition 4.10). 
We must present a piecewise linear function $\phi$ on
$P_n$ such that $\phi$ is linear on all simplices in $\T_n$
and $\phi$ is strictly convex on $P_n$.

It is sufficient to define $\phi$ on vertices of $P_n$.
Let $\phi(0)=0$ and $\phi(\epsilon_{ij})=(i-j)^2$. 
It is not difficult to show that such $\phi$ satisfy the condition of
Definition 4.10.


\enddemo




Now we can complete  the proof of Theorem~2.3.

\demo{Proof of Theorem 2.3}

The first part of Theorem~2.3 is a special case of Theorem~1.4.

The second part follows from Theorems~6.3,~6.4 and Lemma~5.3.
\enddemo

In conclusion of this section we present a construction
of another coherent triangulation of $P_n$.

Let $T$ be a tree on the set $[0,n]$. We say that two edges $(i,j)$
and $(k,l)$ in $T$ are {\it enclosed\/} if $i{<}k{<}l{<}j$.

\definition{Definition 6.5} A tree $T$ on the set $[0,n]$ is
called {\it anti-standard\/} if $T$ is admissible and does not
have enclosed edges. The corresponding base $\I\subset\{(i,j):0{\le} i{<}j{\le} n\}$
is also called {\it anti-standard\/}.
\enddefinition

\proclaim{Theorem 6.6} The set  of anti-standard bases forms a coherent
local triangulation of the polyhedron $P_n$.
\endproclaim

The proof of this theorem is analogous to the proof
of Theorem 6.3. 

\proclaim{Corollary 6.7} The number of anti-standard trees on the set $[0,n]$
is equal to the Catalan number 
$C_n$.
\endproclaim

\medskip








\head 7. Coordinate Strata \endhead


Let $Z_n$ be the group of unipotent matrices $z_{ij}$, 
$ 0{\le} i{\le} j {\le} n$, $z_{ii}=1$
(see Section 2).

Consider a subset $S\subset\{(i,j):0{\le} i{<}j{\le} n\}$.
By $Z_S$ denote the set of all $z=\{z_{ij}\}\in Z_n$
such that $z_{ij}\ne 0$ if and only if $(i,j)\in S$. 
We call $Z_S$
{\it coordinate strata\/} 
in the space $Z_n$. Let $\overline Z_S\simeq\Bbb C^{|S|}$
 be the closure of the stratum
$Z_S$.



We can construct two sheaves of hypergeometric functions
on the manifold $\overline Z_S$, where 
$S\subset\{(i,j):0{\le} i{<}j{\le} n\}$.


First, the  sheaf $\text{Res}_S$
 of restrictions of hypergeometric
functions on $Z_n$ to the manifold $\overline Z_S$.

Second, the sheaf $\text{Sol}_S$ of solutions of the hypergeometric
system (1.2), (1.3) associated with 
$A=A_S=\{e_{ij}:(i,j)\in S\}$ (equivalently, associated with
action (2.1) of torus on $\overline Z_S$).

The question is: when these two sheaves coincide?


\definition{Definition 7.3} Let $\Cal P=\{b_0,b_1,\dots,b_n\}$
be a partially ordered set (poset) such that if
$b_i<_{\Cal P}b_j$ then $i<j$. Consider the set $S_{\Cal P}=\{(i,j):
b_i<_{\Cal P}b_j\}$. We call this set {\it associated\/} with
poset $\Cal P$
\enddefinition



\proclaim{Theorem 7.4} Let $S=S_{\Cal P}$ be the set  
associated with a poset. Then  sheaf $\text{\rm Res}_S$
coincides
with sheaf $\text{\rm Sol}_S$ for generic exponents $\alpha_0,
\dots,\alpha_n$, $\sum\alpha_i=0$.
\endproclaim

\remark{Remark 7.5} By Theorem~1.4 the dimension of
$\text{Sol}_S$ in a neighborhood of a generic point is equal
to $m! \Vol_{H(S)}P(S)$, where $H(S)$ is the 
lattice generated by $e_{ij}$, $(i,j)\in S$, $m=\dim H_S$, and $P(S)$
is the convex hull of the origin and $e_{ij}$, 
$(i,j)\in S$. 
\endremark
\medskip

\proclaim{Proposition 7.6} A set $S\subset\{(i,j):0{\le} i{<}j{\le} n\}$
 is associated with a poset $\Cal P$
if and only if there exists a cone $C$ with vertex at 0 such that
$S=\{(i<j):e_{ij}\in C\}$.
\endproclaim

\demo{Proof}
A set $S$ is associated with a poset
if and only if $S$ satisfies the following transitivity:
if $(i,j), (j,k)\in S$ then $(i,k)\in S$.
The set  $S=\{(i<j):e_{ij}\in C\}$ satisfies transitivity
because if $e_{ij},e_{jk}\in C$ then $e_{ik}=e_{ij}+e_{jk}\in C$.
 Inversely, let $C$ be the cone generated by all $e_{ij}$, $(i,j)\in S$.
If $S$ satisfy transitivity then $S=\{(i<j):e_{ij}\in C\}$.
\enddemo

Now we can prove Theorem 7.4

\demo{Proof of Theorem 7.4} 
Clearly, $\Res_S$ is a subsheaf of $\Sol_S$.
Suppose for simplicity that  $e_{ij}$, $(i,j)\in S$ generate $\Bbb Z^n$.
The dimension of the sheaf $\Sol_S$  at
a generic point is equal
to $n!\Vol(P(S))$ (see Remark 7.5).
Hence, it is sufficient to prove that the dimension of $\Res_S$
at a generic point is greater than  or equal to $n!\Vol(P(S))$.

Let $\T$ be a coherent local triangulation of $P(A)$.
It follows from Proposition 7.6 that $\T$ extends 
to a coherent local triangulation $\T'$ of $P_n$.
Consider  $n!\Vol(P(S))$ $\Gamma$-series $\Phi_\gamma(z)$ on $Z_n$, 
 where $\gamma\in\Pi(\alpha,\I)$, $\I\in\T\subset\T'$.
By Theorem 4.13 these series linearly independent and have common convergence
domain. 
Then restrictions of these series to $\overline Z_S$ give $n!\Vol(P(S))$
independent sections of the sheaf 
$\Res_S$ in some neighborhood.
Therefore, $\Res_S=\Sol_S$.
\enddemo
\medskip




\head 8.  Face Strata  \endhead

 
Describe faces of the polyhedron $P_n$.


Let $I,J\subset [0,n]$, $I\cap J=\emptyset$.
Let $S_{IJ}$ be the set of all $(i,j)$, $ 0{\le} i{<}j{\le} n$
such that $i\in I$ and $j\in J$.

%Let $a=\min I\cup J$ be the minimal element of $I\cup J$ 
%and $b=\max I\cup J$ the maximal 
%element of $J\cup J$.
%We will consider pairs $(I,J)$ such that $a\in I$ and $b\in J$.
% $S_{IJ}\ne\emptyset$ if and only if $a<b$.


\proclaim{Proposition 8.1} Faces $f$ of the polyhedron $P_n$
such that $0\not\in f$ are in 1--1 correspondence with 
%pairs $(I,J)$
%such that $\min I\cup J\in I$ and $\max I\cup J\in J$.
sets $S_{IJ}$.
And  $(i,j)\in S_{IJ}$ whenever  $e_{ij}$ is a vertex of the corresponding
face $f$.
% The set of all $e_{ij}$ such that
% $(i,j)\in S_{IJ}$ is the set of vertices
%of the corresponding face $f$. 
\endproclaim

Clearly, we may assume that $\min(I{\cup}J)\in I$ and $\max(I{\cup}J)\in J$
(if $S_{IJ}$ is nonempty).



Construct a coordinate stratum associated with a face $f$ of $P_n$
$0\not\in f$.

Let $S=S_{IJ}$.
By $Z_{IJ}$ denote the stratum $Z_S$ (see Section 3).
We will call such strata {\it face strata}.


Note that condition (1.5) holds for vectors $e_{ij}$,
$(i,j)\in S_{IJ}$, because all such $e_{ij}$ belong to a
supporting hyperplane
of the corresponding face $f$.

\definition{Definition 8.2} The {\it Hypergeometric System on\/}
$\overline{Z}_{IJ}$ is the hypergeometric system (1.2), (1.3) associated with
the set of vectors $A=\{e_{ij}:(i,j)\in S_{IJ}\}$. Solutions of this system
are called {\it Hypergeometric Functions on\/} $\overline{Z}_{IJ}$.
\enddefinition


\remark{Remark 8.3}
Let  $0{\le}p{<}n$, $I=\{0,1,\dots,p\}$, and $J=\{p{+}1,p{+}2,\dots,n\}$.
Then $\overline{Z}_{IJ}$ is the space of rectangular matrices $z=\{z_{ij}\}$,
$i\in [0,p]$, $j\in [p{+}1,n]$. The hypergeometric system on $\overline{Z}_{IJ}$ is
also called the {\it Hypergeometric System on the Grassmannian\/}
$G_{n{+}1\, p{+}1}$.
This system was studied in the works \cite{GGR1, GGR2, GGR3}.
\endremark

 


It is clear that the set $S=S_{IJ}$ is associated with a poset
 (see Definition 7.3).
Therefore, by Theorem~8.4, the sheaf $\text{Res}_S$ coincides with the sheaf
 $\text{Sol}_S$ of hypergeometric functions on  
$\overline{Z}_{IJ}$ (for generic $\alpha$). 


We will find the dimension of this sheaf in a neighborhood of a generic point.
Denote this dimension by $D_{IJ}$.
In other words, $D_{IJ}$ is the number of independent
solutions of the hypergeometric system on  $\overline{Z}_{IJ}$
 in a neighborhood of a 
generic point.

Let  $P_{IJ}$ be the convex hull of $0$
and $e_{ij}$, $(i,j)\in S_{IJ}$. Let $H_{IJ}$ be
the sublattice generated by $e_{ij}$, $(i,j)\in S_{IJ}$,
and $m=\dim H_{IJ}$.
By Theorem~1.4 the number $D_{IJ}$ is equal to $m!\Vol_{H_{IJ}}(P_{IJ})$.

We present an explicit combinatorial interpretation of this number $D_{IJ}$.


\definition{Definition 8.4}
\roster
\item 
A {\it word} $w$ of {\it type} $(p,q)$ is the sequence 
$w=(w_1,w_2,\dots,w_{p{+}q})$, $w_r\in\{1,0\}$ 
such that 
$|\{r:w_r=0\}|=p$ and $|\{r:w_r=1\}|=q$.


\item
Let  $w=(w_1,w_2,\dots,w_{p{+}q})$ and $w'=(w_1',w_2',\dots,w_{p{+}q}')$
be two words of type $(p,q)$.
We say that $w'$ is {\it exceeds\/} $w$ if $w_1'+\dots+w_r'\ge w_1+\dots+ w_r$
for all $r=1,2,\dots,p{+}q$.
\endroster
\enddefinition

\midinsert
\vskip 10pt
\line{\hfil \epsfysize=3.5cm \epsfbox{figure81.eps}\hfil }
\bigskip
\line{\hfil$w=(0,0,1,1,0,0,1,0,0,0,1,1,0,0,1,0,1)$\hfil}
\line{\hfil$w'=(0,0,1,1,0,0,1,0,0,0,1,1,0,0,1,0,1)$\hfil}
\vskip 10pt
\botcaption{Figure 8.1} The word $w'$ exceeds the word $w$.
\endcaption
\endinsert

We can present a word $w$ of type $(p,q)$ 
as the path $\pi=(\pi_0,\pi_1,\dots,\pi_{p{+}q})$
in $\Bbb Z^2$ such that 
$\pi_s=(i_s,j_s)$ for all $s=0,1,\dots,p{+}q$, 
where $i_s$ (correspondingly, $j_s$) is the number of 0's
(correspondingly, 1's) in $w_1,w_2,\dots,w_s$.
See example for $(p,q)=(10,7)$ on Fig.~8.1.


Clearly, a word $w'$ exceeds a word $w$ if and only if the path $\pi'$ corresponding
to $w'$ is above the path $\pi$ corresponding to $w$.\ 
(See Fig.~8.1.)
 
Let $a=\min I$ and $b=\max J$. Then $D_{IJ}\ne0$ if ond only if $a< b$.

Suppose that $a< b$, $I=\{a\}\cup I'$ and $J=\{b\}\cup J'$, 
where $I',J'\subset [a{+}1,b{-}1]$,
$I'\cap J'=\emptyset$.
 Let $|I'|=p$, $|J'|=q$ and $I'\cup J'=\{t_1{<}t_2{<}\dots<t_{p{+}q}\}$.
Associate with the pair $(I,J)$ the  word $w_{IJ}=(w_1,\dots,w_{p{+}q})$ 
of type $(p,q)$
such that $w_r=0$ if $t_r\in I$ and $w_r=1$ if $t_r\in J$ 
for all $r=1,2,\dots,p{+}q$.


\proclaim{Theorem 8.5} 
The number $D_{IJ}$ is equal to the number of words $w'$ of type
$(p,q)$ which exceed the word $w=w_{IJ}$.
In other words, $D_{IJ}$ is the number of paths $\pi'$ from $(0,0)$
to $(p,q)$ such that $\pi'$ is above the path $\pi=\pi_{IJ}$ corresponding
to $w_{IJ}$.
\endproclaim

\proclaim{Corollary 8.6} Let $I=\{0,2,4,\dots,2k\}$ and
$J=\{1,3,5,\dots,2k{+}1\}$ then $D_{IJ}$ is equal to the Catalan number
$C_k$.
\endproclaim

\demo{Proof}
Words $w'=(w_1',w_2',\dots,w_{2k}')$ of type
$(k,k)$ which exceed the word $w=(1,0,1,0,\dots,1,0)$ are called 
{\it Dyck words}.
 It is well know (see e.g. \cite{SW}) that the Catalan number $C_k$
is equal to the number of 
 Dyck words.
\enddemo
\medskip




\head 9. Standard Triangulation of $P_{IJ}$  \endhead

Let $I,J\subset [0,n]$, $I\cap J=\emptyset$ be two subsets such that
$\min(I{\cup}J)\in I$ and $\max(I{\cup}J)\in J$ (see Section~8).

Recall that   
$P_{IJ}=\Conv(0,e_{ij}:(i,j)\in S_{IJ})$.

In this section we present a coherent local triangulation of the
polyhedron $P_{IJ}$ and prove Theorem~8.5.

\definition{Definition 9.1}
Let $T$ be a tree on the set $I\cup J$. We say that $T$ is  {\it of type\/} 
$(I,J)$ if for every edge $(i,j)$ in $T$ $i\in I$ and $j\in J$.
 The base $\I\subset\{(i,j):0{\le} i{<}j{\le} n\}$ corresponding to $T$ is  
also called  {\it of type} $(I,J)$.
(Do not confuse  $\I$ with $I$.)
\enddefinition

Clearly, all trees of type $(I,J)$ are admissible (see Definition 5.1).


\proclaim{Theorem 9.2} The set $\T_{IJ}$ of all
standard (see Definition~6.1) bases of type $(I,J)$ forms
a coherent local triangulation of the polyhedron $P_{IJ}$.
\endproclaim

The proof of this theorem is essentially the same as the proof of Theorem~6.3.

It is clear that $D_{IJ}=m!\Vol(P_{IJ})$ is equal to the number of
all standard bases (trees) of type $(I,J)$.
Prove that this number coincides with the number given by Theorem~8.5.

\proclaim{Theorem 9.3} Let  $|I|=p{+}1$ and $|J|=q{+}1$.
Then the number of all standard trees $T$
of type $(I,J)$ is equal to the number of words $w'$ of type $(p,q)$
 which exceed the word $w=w_{IJ}$.
\endproclaim

\demo{Proof} 
Let $D_{IJ}$ be the number of all standard trees of type $(I,J)$ and
$\widetilde{D}_{IJ}$ be the number of words $w'$ of type $(p,q)$ which
exceed the word $w=w_{IJ}$ (we use the same notation as in Theorem 8.5).
 
We prove that $D_{IJ}=\widetilde{D}_{IJ}$ 
by induction on $p+q$.
Obviously, this is true for $p=q=0$.

Let $d$ be the minimal element of $J$ and $c$ be the maximal element 
of $I$ such that
$c\le d$.
Let $\widetilde{I}=I\setminus\{c\}$
and $\widetilde{J}=J\setminus\{d\}$.

Prove that if $p+q> 0$ then
$$
D_{IJ}=D_{\widetilde{I}J}+D_{I\widetilde{J}}.
\tag 9.1
$$

  Every standard tree of type $(I,J)$ has the edge $(c,d)$.
In every such tree either $c$ or $d$ is an end-point.
The first choice corresponds to the term $D_{\widetilde{I}J}$
 and the second choice corresponds to the term $D_{I\widetilde{J}}$
in (9.1).

The numbers $\widetilde{D}_{IJ}$ also satisfy the relation (9.1).
The first term corresponds to the case when the word $w'$ starts with 0
and the second term to the case when $w'$ starts with 1.

Therefore, we get by induction $D_{IJ}=\widetilde{D}_{IJ}$.
 \enddemo



Theorem~8.5 is a corollary of Theorem 9.3.
\bigskip




%\head 10. Some Results for Other Root Systems \endhead



\head 10. Examples \endhead

In this and the next sections we present several examples which illustrate
the notions introduced in the paper and show the direction
for following study.

\subhead 10.1. Case $n=2$  \endsubhead

In this case the solutions $f$ of the system (2.2), (2.3)
are functions of variables $z_{01}, z_{02}, z_{12}$.

Let $\beta_1=\frac13(\alpha_2-2\alpha_0)$ and $\beta_2=\frac13(2\alpha_2-
\alpha_0)$.
Because of homogeneous conditions (1.4) we can write
$f(z_{01}, z_{02}, z_{12})=z_{01}^{\beta_1}z_{12}^{\beta_2}\, F(y)$,
where $y=\frac{z_{02}}{z_{01}z_{12}}$.
Now system  (2.2), (2.3) is equivalent to the following equation on $F(y)$.
$$
\frac {dF}{dy}=\(y\frac{d}{dy}-\beta_1\)\,\(y\frac{d}{dy}-\beta_2\)\,F.
\tag 10.1
$$

This is the degenerate hypergeometric equation and its solutions
can be written in terms of the degenerate hypergeometric function
${}_1F_1$ (see \cite{BE}).


This system has two dimensional space of solutions, which is
compatible with the fact that $C_2=2$.
\medskip

\subhead 10.1. Upper triangular matrices \endsubhead

Let $I=\{0,2,\dots,2n\}$ and $J=\{1,3,\dots,2n{+}1\}$.
It is natural to identify the space $\overline{Z}_{IJ}$
with the space of all upper triangular matrices with
arbitrary elements on the diagonal.
Consider the hypergeometric system on $\overline{Z}_{IJ}$.
We call this system {\it the hypergeometric system on upper 
triangular matrices}. 

This system has the same dimension $C_n$ of solution space as 
system (2.2), (2.3) (see Corollary 8.6).
But it is nonconfluent unlikely system (2.2), (2.3).

If fact, system (2.2), (2.3) can be obtained as a limit
of the hypergeometric system on upper triangular matrices.

For example, if $I=\{0,2,4\}$ and $J=\{1,3,5\}$ then the
corresponding hypergeometric system on $\overline{Z}_{IJ}$
can be reduced to the Gauss hypergeometric equation.
And equation (10.1) is a limit of the Gauss hypergeometric equation.















\head 11. Concluding Remarks and Open Problems \endhead

\subhead 11.1. Characteristic manifold\endsubhead


We do not prove here Theorem~1.4. There exist a proof
of this theorem generalizing the proof from  \cite{GZK} for  nonconfluent 
case.

This proof is based on consideration of {\it characteristic manifold\/}
$Ch$ for system (1.2), (1.3).
The characteristic manifold for system (1.2), (1.3) is the submanifold
in the space $\Bbb C^N\times\Bbb C^N$ with
coordinates $(x,\xi)$, $x=(x_1,\dots,x_N),$ $\xi=(\xi_1,\dots,\xi_n)$ 
given by the following algebraic equations.
$$\align
       \sum_{j=1}^N a_{ij}x_j {\xi_j}&=0,
\qquad i=1,2,\dots,n;  \\
     \prod_{j:\,l_j>0}{\xi_j}^{l_j}&=
        \prod_{j:\,l_j<0}{\xi_j}^{-l_j}\qquad\text{if } \sum_j l_j=0;  \\
 \prod_{j:\,l_j>0}{\xi_j}^{l_j}&=0\qquad \text{if }
\sum_{j:\,l_j>0}l_j>\sum_{j:\,l_j<0}l_j,
\endalign
$$
where $l=(l_1,l_2,\dots,l_N)$ ranges over the lattice $L$ of integer
 vectors such that
$l_1a_1+l_2a_2+\dots+l_Na_N=0$.


Then system (1.2), (1.3) is holonomic
if $\dim Ch=N$. The number of independent solutions
at a generic point is equal to  degree of $Ch$ along the
zero section $\{(0,\xi):\xi\in \Bbb C^N\}$ (see \cite{Ka}).
\medskip


\subhead 11.2. Other root systems \endsubhead

 We can define (see Section~2) the hypergeometric system
for arbitrary root system $R$.

It is interesting to find analogues of all results in this paper
for other root systems.

Let $P_{R^+}$ be the convex hull of~0 and all positive roots
$r\in R^+$. Then by Theorem~1.4 the dimension
of the system at a generic point is equal to
$D(R)=n!\Vol(P_{R^+})$, where $n$ is the dimension of $R$.

These numbers $D(R)$ can be viewed as
a generalization of the Catalan numbers for arbitrary root system.

\medskip
\subhead 11.3. Discriminant and Triangulations of $P_n$ \endsubhead






We can associate with system (2.2), (2.3) the discriminant $\cD_n(z)$. 
The discriminant $\cD_n(z)$ is a polynomial of $z=(z_{ij})$, 
$0{\le}i{<}j{\le}n$
such that $\cD_n(z)=0$ if and only if there exists $(z,\xi)\in Ch$ such
that $\xi\ne 0$,
where $Ch$ is the characteristic manifold for system (2.2), (2.3).
%This property defines $\cD_n(z)$ up to a scalar factor.

It is an interesting problem to find an explicit expression for $\cD_n(x)$
and describe all monomials in $\cD_n(x)$.



 The Newton polytope $S_n$ for $\cD_n(x)$ is called 
{\it Secondary polytope}. Vertices of  $S_n$ correspond to
coherent local triangulations of $P_n$ (cf. \cite{GKZ}).

In Section~6 we constructed two coherent local triangulations 
of  $P_n$. The important problem is to find all such triangulations.

Analogously, one can define discriminant $\cD_{IJ}(z)$
associated with face strata $Z_{IJ}$ (see Section~8).
Vertices  of the Newton polyhedron for $\cD_{IJ}(z)$ correspond
to coherent triangulations of $P_{IJ}$.\ (Note that all triangulations
of $P_{IJ}$ are local.)  How  to describe
triangulations of $P_{IJ}$?

The special case of this problem for the pair $(I,J)$ such as in Remark~8.3 
(the hypergeometric system on the  grassmannian)
is connected with
triangulations of the product of two simplices $\Delta^p\times\Delta^q$,
$p+q=n+1$.
In this case $\cD_{IJ}$ is the product of all minors
of $(p{+}1)\times(q{+}1)$-matrix $z$ (see \cite{GKZ}, cf. \cite{SZ, BZ}).
\bigskip



 



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\enddocument