Integer lattice

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In mathematics, The n-dimensional integer lattice, or cubic lattice, denoted Zⁿ, is the lattice in the standard n-dimensional real inner product space, where the inner product is the dot product, and where the lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. Zⁿ is the simplest example of a root lattice.

Automorphism group

The automorphism group of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order

2ⁿ n!.

For the square lattice, this is the group of the square, of order 8; for the three dimensional cubic lattice, we get the group of the cube, or octahedral group, of order 48.

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