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I am a combinatorist by training and I am interested in learning about the connections between combinatorics and Schubert varieties. The theory of Schubert varieties seems to be a difficult area to break into if one has not already studied it in graduate school. I don't have a formal course in Algebraic Geometry, but I do know some Commutative Algebra. Does anyone have any recommendations on what Algebraic Geometry one needs to understand (and what books/papers one should read) in order to begin studying Schubert varieties? Algebraic Geometry is such a huge subject that I am trying to figure out what is essential to this particular study and what is not. Thanks, in advance, for any suggestions!

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7 Answers 7

up vote 16 down vote accepted

I would suggest Part III of Fulton's Young Tableaux book (of which you should skip Part I) as the best starting point for learning about Schubert varieties.

One can get very far in this subject with a naive 19th century view of algebraic geometry, especially if one is willing to occasionally accept without proof a few foundational facts (for example the basics of intersection theory). I would suggest that you don't need to learn algebraic geometry in general for now, though if you're serious about working in this area you'll eventually need to get some feel for what kinds of questions algebraic geometers are interested in.

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I can't second this strongly enough. Fulton's "Young Tableaux" is a beautiful book, and it's my, as an algebraic geometer, only real access into Combinatorics, which I happen to not be terribly talented at. The big downside of Part 3 is that it moves very fast. You might also want to check out a paper by Kleiman and Laksov titled "Schubert Calculus" which appeared in the Notices. –  Charles Siegel Nov 30 '09 at 5:18
    
I think Charles means : SL Kleiman, D Laksov, Schubert Calculus, The American Mathematical Monthly, 1972 , which is indeed a nice survey. –  Georges Elencwajg Nov 30 '09 at 12:49
    
Ahh, that's the one. Couldn't find it quickly and went on memory. –  Charles Siegel Nov 30 '09 at 13:07
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There is one particular fact that greatly helps me understand Schubert varieties and Schubert cells, and the serves as kind-of an introduction: They are a generalization of row echelon form for matrices. If $V \subseteq F^n$ is a $k$-subspace of the standard $n$-space over a field $F$, then it is well-known that $V$ has a unique basis of row vectors in reduced row echelon form. The shape of the form divides the set of all $V$ (the Grassmannian) into cells which are affine spaces. The closures of such a cell — itself plus lower cells — is a Schubert variety. For example, one pattern of RREF for a 2-plane in $F^4$ looks like this: $$\begin{pmatrix} 1 & * & 0 & * \\\\ 0 & 0 & 1 & * \end{pmatrix}$$ This is clearly an affine cell. It is equally easy to show (in the case of a Grassmannian) that the cells are bijective with the $k$-subsets of an $n$-set.

This is not a complete explanation because a Grassmannian is just the simplest type of flag variety, but it captures the basic geometric idea.

As for references, there is an interesting mini-review for combinatorialists on page 398 of Stanley, Enumerative Combinatorics, Vol 2.

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Another reference I've heard suggested before is these notes by Brion:

Lectures on the geometry of flag varieties

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I second the recommendation of Fulton's book. Another good source is Chapters 14-16 of Combinatorial Commutative Algebra, by Miller and Sturmfels. You don't need to read the chapters before them! Just jump in and refer back as needed.

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There is a new book by Lakshmibai and Raghavan called Standard Monomial Theory which is mostly about how to do invariant theory in a "Schubert varietiesque" way. It is introductory to both Schubert varieties and invariant theory. I haven't read all of it, but I would recommend it because it works out a lot of different cases. I don't think you need that much algebraic geometry to read it. They review the relevant facts beforehand.

edit: I should also mention that one advantage of this book over Fulton (which I also recommend) is that it covers Schubert varieties in other types of Grassmannians. Also, it gives a nice application of Schubert varieties to invariant theory (in particular, how to calculate rings of invariants in a characterstic-free way in various cases of classical interest).

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I'd also recommend Part III of Fulton's "Young tableaux". Also, if you want a really basic introduction (with some technicalities skipped) to some Schubert calculus and intersection theory you could try reading Sheldon Katz's "Enumerative Geometry and String Theory" which are lecture notes from a Park City advanced undergraduate course. In chapter 7 you can find Schubert calculus of G(2,4), i.e., planes in four dimensional complex space.

Here's an AMS page for the book.

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Katz also talks a bit through $G(2,n)$, though he's only really interested in curve counting, so you get lines, and then he moves on to quantum cohomology related topics, so only a small section is useful. –  Charles Siegel Nov 30 '09 at 13:08
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Not exactly about Schubert Varieties, but worth mentioning:

This article by Henry Cohn gives a very nice point of view on some combinatorical questions as being questions about geometry over "the field with on element". Every Schubert cell of a Schubert variety should contain exactly one F_1-valued point...

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