Difference between revisions of "Unimodular lattice"
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| + | $#C+1 = 9 : ~/encyclopedia/old_files/data/U095/U.0905360 Unimodular lattice | ||
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| + | 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==== | ====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=18539
Unimodular lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_lattice&oldid=18539