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?
Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$
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reference-request
representation-theory
integer-lattices
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1It 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.) – 2012-06-27
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0I'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. – 2012-06-27
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0Relationship to what and applications to what? – 2012-06-27
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0You 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). – 2012-06-27
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0Applications 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. – 2012-06-27
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0Reiner's Maximal Orders discusses arithmetic aspects of this, using complete local rings (as easy first steps to handle the harder questions over Z). I don't think it does any function field stuff. You might it handy if your lattices carry a non-commutative ring structure on them. Algebraic number theory books are probably useful if they carry commutative ring structures (though again, probably a focus on Z rather than C[t]). – 2012-06-27
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0I think Serre's Lie Groups and Lie Algebras has a discussion of generalities on O-lattices in a finite-dimension F-vector space, when O is a complete DVR and F is its fraction field (such as your ${\mathbf C}[[t]]$, or ${\mathbf Z}_p$). – 2012-06-27