- #1

Peter_Newman

- 145

- 10

Hello,

I want to prove that for any lattice ##\Lambda = \Lambda(B)## (the ##B## is a basis) the orthogonalized parallelepiped ##P(B^*) = B^*\left[-0.5,0.5\right)^n## is a fundamental region of a lattice.

If I wanted to show this, I would try to establish a volume argument here. After all, the determinant of a lattice is no more than the volume of the fundamental region, by definition. Now I already know that ##det(\Lambda) = |det(B)| = |det(B^*U)| = \prod ||b_i^*||##. The ##B = B^{*}U## is just the Gram-Schmidt composition as matrix, ##B^*## is an orthogonal matrix and ##U## is an upper triangular matrix containing the Gram-Schmidt coefficients. So now we go on, by asking what is the volume of ##P(B^*)##, that is the determinant ##det(B^*)## and that is just the product again ##\prod ||b_i^*||##. So here we have a direct relationship between ##vol(P(B))## and ##vol(P(B^*))##. If we already know that ##P(B)## is a fundamental region, then we can conclude from the volume argument that ##P(B^*)## is also a fundamental region.

This raises two questions for me. First, is this meant as a "proof" in the right way? Second, can we formalize this a little better, make it a little more concrete?

I want to prove that for any lattice ##\Lambda = \Lambda(B)## (the ##B## is a basis) the orthogonalized parallelepiped ##P(B^*) = B^*\left[-0.5,0.5\right)^n## is a fundamental region of a lattice.

If I wanted to show this, I would try to establish a volume argument here. After all, the determinant of a lattice is no more than the volume of the fundamental region, by definition. Now I already know that ##det(\Lambda) = |det(B)| = |det(B^*U)| = \prod ||b_i^*||##. The ##B = B^{*}U## is just the Gram-Schmidt composition as matrix, ##B^*## is an orthogonal matrix and ##U## is an upper triangular matrix containing the Gram-Schmidt coefficients. So now we go on, by asking what is the volume of ##P(B^*)##, that is the determinant ##det(B^*)## and that is just the product again ##\prod ||b_i^*||##. So here we have a direct relationship between ##vol(P(B))## and ##vol(P(B^*))##. If we already know that ##P(B)## is a fundamental region, then we can conclude from the volume argument that ##P(B^*)## is also a fundamental region.

This raises two questions for me. First, is this meant as a "proof" in the right way? Second, can we formalize this a little better, make it a little more concrete?

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