View source for Higher spin alternating sign matrix

{{Refimprove|date=November 2009}}

In [[mathematics]], a '''higher spin alternating sign matrix''' is a generalisation of the [[alternating sign matrix]] ('''ASM'''), where the columns and rows sum to an integer ''r'' (the ''spin'') rather than simply summing to 1 as in the usual alternating sign matrix definition. HSASMs are square matrices whose elements may be integers in the range &minus;''r'' to +''r''. When traversing any row or column of an ASM or HSASM, the partial sum of its entries must always be non-negative.<ref name="ecomjournal">[http://www.combinatorics.org/Volume_14/PDF/v14i1r83.pdf R. E. Behrend and V. A. Knight, "Higher Spin Alternating Sign Matrices", ""The Electronic Journal of Combinatorics"", '''14''' (2007), #R83]</ref>

High spin ASMs have found application in [[statistical mechanics]] and [[physics]], where they have been found to represent [[symmetry group]]s in [[ice crystal]] formation.

Some typical examples of HSASMs are shown below:

:<math>
\begin{pmatrix}
 0 & 0 & 2 & 0 \\
 0 & 2 &-1 & 1 \\
 2 &-1 & 2 &-1 \\
 0 & 1 &-1 & 2 
\end{pmatrix};\quad
\begin{pmatrix}
 0 & 0 & 2 & 0&0 \\
 0 & 1 &-1 & 2 &0\\
 2 &-1 &-1 & 0 &2\\
 0 & 0 & 2 & 0 &0\\
0&2&0&0&0
\end{pmatrix};\quad
\begin{pmatrix}
 0 & 0 & 0 & 2 \\
 0 & 2 & 0 & 0 \\
 2 &-2 & 2 & 0 \\
 0 & 2 & 0 & 0 
\end{pmatrix};\quad
\begin{pmatrix}
 0 & 2 & 0 & 0 \\
 0 & 0 & 0 & 2 \\
 2 & 0 & 0 & 0 \\
 0 & 0 & 2 & 0 
\end{pmatrix}.
</math>

The set of HSASMs is a [[superset]] of the ASMs. The [[extreme points]] of the [[convex hull]] of the set of ''r''-spin HSASMs are themselves integer multiples of the usual ASMs.

==References==
<references />

{{DEFAULTSORT:Higher Spin Alternating Sign Matrix}}
[[Category:Matrices]]
[[Category:Statistical mechanics]]
[[Category:Enumerative combinatorics]]


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