# Infinite Grassmannians and their coordinate rings

I'm currently thinking about some combinatorics associated to an infinite analogue of the coordinate rings of the Grassmannians $Gr(2,n)$. The combinatorics should be thought of as relating to Plucker coordinates $\Delta^{ij}$ but with $i < j$ arbitrary integers, rather than restricted to $\{1,\ldots ,n\}$. So I've been trying to find the right infinite Grassmannian to have this coordinate ring (or, if the regular functions are a bit more complicated in the infinite case, to at least have these Plucker coordinates in there). I've looked (briefly) into:

(a) Kac's construction of infinite Grassmannians ([Kac, Infinite dimensional Lie algebras, 3rd ed.], Exercise 14.32, p.339)

(b) taking the union of the finite Grassmannians to get a classifying space for $O(n)$ or $U(n)$

(c) infinite Grassmannians coming from Hilbert spaces ([Pressley and Segal, Loop groups])

but none of these seem to quite describe what I want. Some of these are working with $\mathbb{N}$-dimensional space rather than $\mathbb{Z}$-dimensional space (i.e. something more like $\mathbb{C}[t]$ than $\mathbb{C}[t,t^{-1}]$), usually from a colimit of the finite ones, and I can't see how to alter the definition and be sure of keeping the theorems. And with the others that do work with something like $\mathbb{C}[t,t^{-1}]$, I can't see a description that corresponds to planes in that space (and I definitely need just the planes).

I feel sure this is well-known so does anybody know a reference for both the construction I want and also enough information about its coordinate ring?

Edit: Having thought about this a little more, I want to formulate the question more specifically as:

Let $V=\mathbb{C}[t,t^{-1}]$ and define $Gr(2,V)$ to be the set of 2-dimensional subspaces of $V$. Does the finite-dimensional machinery of the Plucker embedding work in this setting and give Plucker coordinates in $\mathbb{C}[Gr(2,V)]$ of the form $\Delta^{ij}$ for integers $i < j$?

## 1 Answer

I have put a brief note that might be helpful at

http://neil-strickland.staff.shef.ac.uk/research/fock.pdf

• @Neil: thanks for this link. Perhaps it's that I'm not very expert in this but I can't quite see what I want in your notes. Would you be able to give me a pointer to the exact part I should look at it more detail? Or even give a brief outline sketch of the idea here? Thanks! – Jan Grabowski Mar 2 '11 at 11:48
• Jan: it seems that Neil is referring to the opening paragraph together with definition 1.1 (Neil, feel free to correct me if I'm wrong). – Adam Hughes Mar 2 '11 at 15:17
• ^ I mean that in the sense that that might be the way Neil suggests you should think about infinite dimensional Grassmannians, the rest of the note seems to give how well this notion restricts to the finite dimensional case, which might be helpful in "theorem preservation" you noted concern with in your original question. – Adam Hughes Mar 2 '11 at 15:21