\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Roger E. Behrend and Vincent A. Knight}
%
%
\medskip
\noindent
%
%
{\bf Higher Spin Alternating Sign Matrices}
%
%
\vskip 5mm
\noindent
%
%
%
%
We define a higher spin alternating sign matrix to be an integer-entry
square matrix in which, for a nonnegative integer $r$, all complete
row and column sums are $r$, and all partial row and column sums
extending from each end of the row or column are nonnegative. Such
matrices correspond to configurations of spin $r/2$ statistical
mechanical vertex models with domain-wall boundary conditions. The
case $r=1$ gives standard alternating sign matrices, while the case in
which all matrix entries are nonnegative gives semimagic squares. We
show that the higher spin alternating sign matrices of size~$n$ are
the integer points of the $r$-th dilate of an integral convex polytope
of dimension $(n{-}1)^2$ whose vertices are the standard alternating
sign matrices of size~$n$. It then follows that, for fixed $n$, these
matrices are enumerated by an Ehrhart polynomial in $r$.
\bye