In the proof that $SL_{n}(\mathbb{Z})$ is a lattice in $SL_{n}(\mathbb{R})$, the following isomorphism is used $$SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z}) \cong \{\text{unimodular lattices in } \mathbb{R}^n \text{ with covolume 1} \}$$
(for example, See proof of theorem 17 in section 2.2.1 here)
The covolume of a (full-rank) lattice in $\mathbb{R}^n$ is $|\text{det}B|$ of its basis $B$. This is well-defined, since any two bases of the same lattice are related by a unimodular transformation (with determinant $+1$ or $-1$).
Now I am confused, because a lattice having covolume $1$ doesn't imply that it has a basis in $SL_{n}(\mathbb{R})$, it's possible that it only has a negatively-oriented basis with determinant $-1$ instead. So the "isomorphism" doesn't seem to be surjective. Another thing, why take the quotient with $SL_{n}(\mathbb{Z})$ when it's possible that two bases of the same lattice are related by unimodular matrix with determinant $-1$, this still would preserve the covolume since we are taking the absolute value.
Is the definition of the covolume incorrect? It doesn't care about the orientation, while the matrix spaces are specifically orientation preserving..
Any clarification would be great!
Best Answer
Given a basis $B$ for a unimodular lattice with $\det(B)=-1$, you can just negate one of the terms of $B$ to get a basis $B'$ for the same lattice with $\det(B')=1$. So, every unimodular lattice can be represented by a basis in $SL_n(\mathbb{R})$. Moreover, given two elements of $SL_n(\mathbb{R})$ that represent the same unimodular lattice, the change of basis matrix between them is in $SL_n(\mathbb{Z})$ (since it is a composition of two matrices of determinant $1$). So, the set of unimodular lattices is in bijection with the coset space $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$.
(It is also in bijection with $SL^\pm_n(\mathbb{R})/SL^{\pm}_n(\mathbb{Z})=SL^\pm_n(\mathbb{R})/GL_n(\mathbb{Z})$ where $SL^\pm$ denotes the group of matrices with determinant $\pm 1$, since you could instead consider arbitrary bases with determinant $\pm 1$ and then the change of basis between two such bases could have determinant either $1$ or $-1$.)