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Difference between revisions of "Unimodular lattice"

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A lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953601.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953602.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953603.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953604.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953605.png" /> vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953606.png" />, then the lattice spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953607.png" /> is unimodular if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953608.png" /> (because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095360/u0953609.png" />).
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A lattice  $  L $
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in  $  \mathbf R  ^ {n} $
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such that $  \mathop{\rm vol} ( \mathbf R  ^ {n} \mid  L) = 1 $.  
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If $  a _ {1} \dots a _ {n} $
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are $  n $
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vectors in $  \mathbf R  ^ {n} $,  
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then the lattice spanned by $  a _ {1} \dots a _ {n} $
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is unimodular if and only if $  |  \mathop{\rm det} ( a _ {1} \dots a _ {n} ) | = 1 $(
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because $  \mathop{\rm vol} ( \mathbf R  ^ {n} \mid  L ( a _ {1} \dots a _ {n} )) = |  \mathop{\rm det} ( a _ {1} \dots a _ {n} ) | $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  D. Husemoller,  "Symmetric bilinear forms" , Springer  (1973)  pp. 16</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  D. Husemoller,  "Symmetric bilinear forms" , Springer  (1973)  pp. 16</TD></TR></table>

Latest revision as of 08:27, 6 June 2020


A lattice $ L $ in $ \mathbf R ^ {n} $ such that $ \mathop{\rm vol} ( \mathbf R ^ {n} \mid L) = 1 $. If $ a _ {1} \dots a _ {n} $ are $ n $ vectors in $ \mathbf R ^ {n} $, then the lattice spanned by $ a _ {1} \dots a _ {n} $ is unimodular if and only if $ | \mathop{\rm det} ( a _ {1} \dots a _ {n} ) | = 1 $( because $ \mathop{\rm vol} ( \mathbf R ^ {n} \mid L ( a _ {1} \dots a _ {n} )) = | \mathop{\rm det} ( a _ {1} \dots a _ {n} ) | $).

References

[a1] J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) pp. 16
How to Cite This Entry:
Unimodular lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_lattice&oldid=49078