# Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring of integers (in my case, $\mathbb{C}[[t]]$), but I haven't been able to find any references or complete exposition. I believe the bijection is given by sending $x \in \operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ to $x\Lambda$, where $\Lambda$ is the standard lattice in $O$ (so, I think, just $O$ itself). Does anyone know of a book or paper that might have more detail?

-
It suffices to show that $\text{GL}_n(F)$ acts transitively and that $\text{GL}_n(O)$ is the stabilizer. Which step are you having trouble with? (Note that a general field does not have a unique "ring of integers," and in general it is unclear to me what the correct definition of "lattice in $F^n$" should be unless $F$ has more structure.) –  Qiaochu Yuan Jun 27 '12 at 19:17
I'm not having trouble with the bijection, per se- I haven't written it down, but I believe it- but I need to use these lattices to solve some problems about subspaces of $GL_n(F)/GL_n(O)$, so I was hoping to find a resource that fleshes out the relationship and applications. –  Pine Lamb Jun 27 '12 at 19:22
Relationship to what and applications to what? –  Qiaochu Yuan Jun 27 '12 at 19:57
You have a typo: "where $\Lambda$ is the standard lattice in $O^n$ (so $O^n$ itself)." I don't know of a good exposition of the relationship (most will start with "we like lattices, did you know GL acts on them?" not "we like GL/GL, did you know lattices are involved?"). I think it is more common to see this with F the rational numbers and O the integers. At any rate, F^n is an injective O-module, so any embedding of O^n in F^n extends to a endomorphism of F^n that sends the original O^n to the new O^n. Since the image spans F^n, the endomorphism is in GL(n,F). –  Jack Schmidt Jun 27 '12 at 20:54
Applications I usually see: we have lattices with extra structure (group action), and we want to classify them up to O-equivalence, but that is hard at first, so we do it up to F-equivalence first, and then break each F-equivalence class apart. Crystallography and modular rep theory of finite groups work this way. –  Jack Schmidt Jun 27 '12 at 20:56