I got a bit behind on responding to comments to my post on Li’s preprint, so I thought I would just start a new thread.

Now, I don’t want to concentrate too much on the particular’s of Li’s case, since I don’t know Li or too many of the specifics.  I’ll just clarify that I don’t think Li is a crackpot, or did I say in my past post that I though that.  But that’s exactly why I feel like releasing his preprint the way he did was a mistake.  Certainly, there’s a bit of my taste for facetious exaggeration (I’m sure my readers have noticed this tendency in the past) in describing the unfortunate aspects of that as “crackpotesque,” but I think that also helps convey what about it I thought was a bad decision.

I’m more interested in expanding on some of the comments in that thread.  Gil Kalai said:

There are two extreme ways to practice math (with many altenatives in between.) One way is to work secretly on a big problem, to tell nobody or very few people about it, to discuss with nobody the techniques you are using, and then after many years to astonish the world with a preprint or a lecture) presenting the solution. The other extreme way is to work while at any time discussing your thoughts and ideas with everbody (perhaps also on blogs), write papers with partial progress and conjectures etc.

The advantage of the first avenue is not just the fear that somebody will use your ideas but also that it helps the researcher to stay concentrated, and avoid outside preasure and distractions of various types. A clear disadvantage of the first avenue is that feedbacks from others can be useful at intermediate stages of the process towards a mathematical discovery.

I’m curious: does anyone out there think that Gil’s “first avenue” sounds like a good idea?  It sounds crazy to me.  Maybe I lack the self-confidence to think I would succeed at it (not something I’m regularly accused of), but it seems like asking for trouble, both in terms of actually getting the math done and in terms of one’s career.  Obviously, there are dangers in revealing your ideas and results to other people.  I think outright theft is relatively rare, but someone “eating your lunch,” implementing something you had hoped to do before you have a chance, is a very serious concern.

But I think people’s cognitive biases cause them to be too sensitive to this possibility, while forgetting about the upside, because the danger of having one’s work stolen is so obvious and painful, and the dangers of secrecy are much less obvious.  It’s important to remember though, a co-authored paper which actually happens is much better than a solo one which never does, or even which happens a few years down the road (at least for those of us worried about jobs).  Not to mention the very real possibility that people will independently come up with the results you wanted.  As Greg said:

What is true is that you’re much more likely to lose credit by being secretive than by being open.

I find it interesting that Gil mentions that

The first avenue, had spectacular successes in the last few decades…

I assume he’s referring to Wiles’s proof of Fermat and Perelman’s proof of Poincare (are there other examples I’m missing?), which were certainly both spectacular, but they were somewhat qualified as successes.  I mean, Wiles’ worked mostly in secret for years, announced a false proof, and then fixed it after getting input from other people.  It’s likely that he would have discovered his mistake much earlier if he had been talking to a larger circle of people; one could argue that the level of secrecy he maintained might well have cost him the Fields Medal, which he would have been a shoo-in for if his initial proof had been correct (he had turned 40 by the time he fixed it).

In the case of Perelman, well, what can one say?  One could argue that his behavior has seriously compromised his standing in the mathematical community, but I suppose he doesn’t really care.  Certainly his work would have been a lot more digestable if he had consulted with people more, and seems to be sufficiently complicated that it would have been awfully hard for someone to absorb quickly enough to have scooped him.

Not to mention that this style is much less suited to what most mathematicians do than, say, Wiles’s situation.  When it’s very important for you to prove a particular result which many other people would like to prove, secrecy makes a certain amount of sense, but when you want to establish a reputation as an effective and interesting researcher, having other people build on your research is the best thing that could happen (even if they’re solving problems you had hoped to do yourself), given that it happens pretty rarely.  You’ll be a much better mathematician if you think of people interested in the same problems as potential coauthors rather than potential rivals.

As both Greg and Terry mentioned, having coauthors is a very good thing on a lot of levels.  Obviously, they can stress you out from time to time, but as a general rule, they will lead to you doing better and more mathematics, and writing better and more papers.  On some level, the most important thing to remember is that like trade, talking to other people about mathematics is very much a positive sum game, since the other person is likely to have some small piece of knowledge that you are lacking, or to bring some skill or temperment to a collaboration.  I’m by nature a very impetuous, big-picture-oriented research, and having someone who focuses more on details and makes me come back to earth is a large boon. There’s some risk that like the gains from trade, gains from mathematical discussion will be inequitably distributed, but they’re still indisputably there.

So, Charles over at Rigorous Trivialities recently put up a post begging for submissions to the latest installment of the Carnival of Mathematics, and this prompted Ben and I to wonder:  Why is it customary to request permission to link to posts for the Carnival of Mathematics?  [Edit:  Is it customary?] Is this just standard operating procedure for blog carnivals?

It seems a bit bizarre.  We don’t ask permission to link to posts in any other circumstance.  And it makes it harder to assemble a Carnival.  Obviously, it would be best if people actually submitted their own posts, but I don’t see much sense in not linking to a post because you don’t have explicit permission.  (Ben, I think, didn’t bother to ask anyone for permission when he assembled the SBS-hosted Carnival.)

I suppose some folks might not be comfortable with not having explicit permission.  Maybe we can do something about that in this post.  I, for one, want Carnival hosters to know:  Any post I write is fair game for the Carnival of Mathematics.  If you feel the same way, you might mention it in the comments.

If you ever need an example of how unhelpful and badly designed our current publishing system is, the existence (or rather, lack of existence) of out-of-print books is ready-made.

Now there was a time when not publishing a book could make serious economic sense.  Publishers couldn’t afford to publish runs of books below a certain number, and the demand for some books can become so small that there was no way to profitably print them.  It’s a shame but an understandable economic reality.

This is simply no longer the case.  Print-on-demand services (for example, lulu.com) can now print books as people order them for a cost considerably lower than the list price of any math textbook.  All a publisher needs to do is put PDFs of their books on such a website, put a $30 markup on them (or more, considering how much math books cost), and let the money roll in.  If they don’t have PDFs, I bet Google Books would make them for free.  In short, publishers are leaving money they could be making on their back catalogue on the table, and hurting the mathematical community at the same time.  Thanks, guys.

This rant was engendered by a post of Timothy Chow’s at What’s New (a.k.a. Terry Tao) about a new website, where one can express one’s desire for a old math books to be brought back into print.  The website’s a good idea but ultimately getting specific books that are particularly popular back into print is a short-term fix.  The real problem is that publishers’ mindset still hasn’t caught up to the advances in technology. When are they going to enter the 21st century?

[Ed. - last paragraph edited a bit in response to comments]

There was a request containing the phrase, “theory of modular forms,” so I’ll write an introduction to that. Chris seems to be taking care of the rest of that paragraph.

Pretty much all of the material below is 50-150 years old. Don’t expect too much originality.

For the duration of this post, an elliptic curve will be a complex manifold isomorphic to , where is a discrete subgroup of the complex numbers. An elliptic curve then has the topology of a 2-torus, and the structure of the abelian group U(1) x U(1). However, the complex structure depends nontrivially on the choice of lattice. In particular, two elliptic curves are isomorphic if and only if there is some nonzero complex number r such that . This means we can rotate and dilate lattices without changing the curve, but that’s all. This can be proved using Weierstrass’s theory of meromorphic functions.

Let’s try to classify elliptic curves. We can choose an oriented basis of the lattice (i.e., the first generator is less than 180 degrees clockwise from the second generator), and then rescale the lattice (by rotating and dilating) so that the first generator is equal to one. The second generator is then a point in the complex upper half plane H. The fact that we chose an oriented basis means that H doesn’t classify elliptic curves, since we added extra structure (in particular, it classifies elliptic curves with an oriented basis for first homology). However, the group of two-by-two integer matrices with determinant one acts simply transitively on all such oriented bases, so elliptic curves are classified by taking the quotient of the upper half plane by a certain action of .

It is a fairly well-known fact (which I won’t prove here) that is generated by and . If we have a lattice with oriented bases (1,z), then T fixes 1 and sends z to z+1, yielding the new basis (1,z+1). S takes 1 to z and z to -1, so we divide this new basis by z to get (1,-1/z). More generally, acts on H via , but we can get the structure of the quotient from just the generators. Since T acts by Translation by one, we can choose orbit representatives in the part of H that lies in a vertical strip of width one. S acts by Spinning around i, switching the interior of the unit disc with the exterior. The standard fundamental domain for the action is then the part of the upper half plane outside the unit disc, and with real part between -1/2 and 1/2. You can see a picture of it here. To take the quotient, we glue the left and right sides of the domain together to get an infinitely long tube, and then we glue the bottom shut. This gives us a complex analytic space that is topologically (in fact, complex analytically) a plane, and points in this space classify elliptic curves up to isomorphism. We will call this space Y(1). If we compactify by adding a point at infinity (called a cusp), we get a sphere called X(1), which classifies “generalized elliptic curves.” The extra point describes what you get by taking a sphere and identifying two points so they intersect transversely. There is a group structure on the smooth locus, isomorphic to , so we have essentially let the second generator of our lattice run away to infinity.

(Advanced bit: doesn’t act freely on the half plane, i.e., there are nonidentity elements that fix points, and these fixed points correspond to curves with automorphisms. In particular, every elliptic curve has a -1 automorphism, and the square and triangular lattices have automorphisms of order 4 and 6, respectively. If we want to produce a universal family over a moduli space, we will have to use the machinery of stacks. Deligne and Rapoport showed that the functor Y(3) producing elliptic curves together with an identification of their three-torsion is representable in schemes, so Y(1) is a quotient by the order 24 group . If we look at curves in characteristic 2, there is one elliptic curve whose automorphism group is exactly this one. Clearly, the lattice picture doesn’t work here, since the group is nonabelian. In fact, it is naturally the group of units in a certain quaternion algebra of endomorphisms of the curve.)

So, what is a modular form? I won’t give an answer yet, but a modular function is just a complex function on the upper half plane that is invariant under the action of . Equivalently, it is a function on Y(1), or an invariant of elliptic curves. There is a distinguished subspace of these functions given by those that classify elliptic curves uniquely. If we look at Y(1), these are just one-to-one functions. We typically ask for modular functions to be reasonably nice, i.e., holomorphic, and with reasonable growth as z tends toward infinity. The conditions imply the corresponding function on Y(1) (viewed as a plane) is a polynomial. The one-to-one functions then have the form aj+b for some function j, where a is nonzero. The function j is periodic and holomorphic on the upper half plane, so its Fourier expansion (which I will describe later) has constant coefficients. With a good choice of normalization, the coefficients are nonnegative integers, and this might lead you to suspect that there is some interesting graded vector space whose dimensions are given by these coefficients. This is part of “moonshine.”

We are looking for forms rather than functions, so let’s consider the differential 1-form dz on the upper half plane. If we transform the half-plane by , we get . In other words, if some function f satisfies , then f(z)dz is a one-form that is invariant under , i.e., it lives on the quotient Y(1). Such a function f is called a modular form of weight 2. If f satisfies for all , then f is called a modular form of weight 2k. In general, these forms will not be differential k-forms (i.e., sections of ), but they will describe sections of the pluricanonical bundle , which has the advantage of being nonzero for lots of k. Earlier, we gave an interpretation of modular functions as invariants of elliptic curves. A modular form of weight 2k is an invariant of a pair , where E is an elliptic curve, and is a nowhere-vanishing differential on E (such as dz - there is only a worth of these), and it satisfies . We write f the function to denote the form rather than , because we can trivialize the pluricanonical bundle on the upper half plane by forgetting dz. As the calculation above shows, this trivialization is not -equivariant. There are no nonzero forms of odd weight, because the matrix fixes points and acts by minus one on functions.

Let’s try to write down some examples of forms. A good first place to look is functions of lattices. Since these lattices live in the complex numbers, we can multiply and add, so we consider the function . The factor of two is to prevent cancellation, and we ask that k be greater than one to make this sum converge absolutely. It is not invariant under dilation or rotation, but the nonzero complex numbers act through -2k powers. If we restrict to lattices generated by (1,z), we get a holomorphic function on the upper half plane. It is invariant under T, and for S, , so it is indeed a weight 2k modular form. If we send z to infinity, then all of the non-integer contributions in the lattice sum go to zero, and we are left with as the constant term of the Fourier expansion. In particular, these forms are holomorphic on X(1). One often normalizes them so that the Fourier expansion has constant term 1, and then they are called the Eisenstein series of weight 2k. They have Fourier expansion , where B denotes Bernoulli numbers (which are rational), is the sum of the (2k-1)st powers of all divisors of n, and is a coordinate on the unit disc.

We can multiply Eisenstein series together to get forms of other weights that are not necessarily Eisenstein series, and in fact, the graded ring of modular forms that are holomorphic on X(1) is a polynomial ring generated by and , which are algebraically independent. It is easy to check that there are no forms of odd weight, since we pick up a minus sign when we square S. There are several ways to determine the dimension of the space of forms of a given weight (e.g., orbifold Riemann-Roch), and the fact that the spaces of forms of weight 4,6,8,10,and 14 have dimension 1 implies relations like , which in turn give identities like . There is a two-dimensional space of weight 12 forms, spanned by and . The difference is a form , whose Fourier expansion has no constant term, so it is called a cusp form. is called the discriminant, since it vanishes exactly when a plane cubic is singular. In particular, it doesn’t vanish on the upper half plane, so multiplication by produces an isomorphism between modular forms of weight 2k and cusp forms of weight 2k+12. Also, the quotient is holomorphic of weight zero on Y(1), with a pole at infinity. j has Fourier expansion . The coefficients of these forms satisfy lots of interesting congruence properties, and this is more or less where the theory of p-adic modular forms takes off.

You might be wondering about the use of the term “weight” above. Usually in mathematics, a weight is a representation of a torus, and modular forms are no exception. Here, the torus in question is a maximal compact subgroup . We will write G for the big group, and K for the compact. Iwasawa decomposition splits G as NAK, where and . The group B = NA acts transitively on H, since , and this identifies H with G/K (i.e., G forms a circle bundle over the upper half plane). Elements of G can be written as a point in H together with an angle, and the matrix is taken to . Given a modular form f of weight 2k, we can then produce a function F on G by . F is naturally left invariant under , and K acts on the right by . There is a right regular action of G on any reasonable space of functions on G, and one can actually characterize modular forms f on H as those that correspond to certain eigenfunctions F of the Laplacian (aka Casimir) on G satisfying additional analytic conditions. Modular forms then describe lowest weight vectors for certain (infinite dimensional) unitary representations of G known as discrete series. The raising and lowering in these representations is given by first order differential operators, and the annihilation of the lowest weight vector by a lowering operator is equivalent to the fact that the modular forms satisfy the Cauchy-Riemann equations, i.e., they are holomorphic on H.

So, in past posts, I’ve attempted to explain a bit about Gale duality and about Koszul duality, so now I feel like I should try to explain what they have to do with each other, since I (and some other people) just posted a preprint called “Gale duality and Koszul duality” to the arXiv.

The short version is this: we describe a way of getting a category  (or equivalently, an algebra) from a linear program (or as we call it, a polarized hyperplane arrangement).

Before describing the construction of this category, let me tell you some of the properties that make it appealing.

Theorem. is Koszul (that is, it can be given a grading for which the induced grading on the Ext-algebra of the simples matches the homological grading).

In fact, this category satisfies a somewhat stronger property: it is standard Koszul (as defined by Ágoston, Dlab and Lukács.  Those of you with Springer access can get the paper here).  In short, the category has a special set of objects called “standard modules” (which you should think of as analogous to Verma modules) which make it a “highest weight category,”  such that these modules are sent by Koszul duality to a set of standards for the Koszul dual.

Of course, whenever confronted with a Koszul category, we immediately ask ourselves what its Koszul dual is.  In our case, there is a rather nice answer.

Theorem. The Koszul dual to is , the category associated to the Gale dual of .

Now, part of the data of a linear program is an “objective function” (which we’ll denote by ) and of bounds for the contraints (which will be encoded by a vector ).  Stripping these way, we end up with a vector arrangement, simply a choice of a set of vectors in a vector space, which will specify the constraints.

Theorem. If two linear programs have same underlying vector arrangment, the categories may not be equivalent, but they will be derived equivalent, that is, their bounded derived categories will be equivalent.

Interestingly, these equivalences are far from being canonical. In the course of their construction, one actually obtains a large group of auto-equivalences acting on the derived category of , which we conjecture to include the fundamental group of the space of generic choices of objective function.

While we hope to present some more motivated definitions in the future, let’s start with an explicit presentation of our algebra, just so we can see it’s nothing scary.

Remember that a linear program consists of a real vector space , a collection of n linear functions , and a choice a pair of vectors .

The dual of a program replaces with its perpendicular under dot product in , and .

We consider a sign vectors .  We let and .

We call bounded if for each , we have .

We call feasible if the set is non-empty.

As we discussed, the duality theorem for linear programming implies that a sign vector is feasible for one program if and only if it is bounded for the dual.  In particular, the set of sign vectors which are bounded and feasible for and for coincide.  We will call this set , and it will index the simple modules of our algebra.

Now, consider the quiver whose vertices are the set of feasible sign vectors for , with an edge connecting any two vectors which differ in a single place.  More visually, one can think of the vertices as attached to the chambers , and the edges as connecting chambers adjacent across a single hyperplane.  We consider the path algebra of this quiver tensored with the algebra of polynomial functions on , .  Let be the “lazy path” of length 0, which stays at , and if and are feasible sign vectors which only differ in the ith place is the path of length of two which begins at , changes the ith coordinate (i.e. crosses the ith hyperplane) to  and goes back to .  The algebra of interest to us, which we will denote , is a finite dimensional quotient of this path algebra, by the relations

• any two paths from to of minimal length (i.e. one where the length is the same as the number of coordinates where the vectors differ) are equal.
• for each and each hyperplane adjacent to , we have , where is considered as an polynomial on .
• for each unbounded , we have , i.e. any path passing through that vertex is 0.

The reader may wonder why we included the infeasible vertices at all, if we were going to just declare them to be 0; the answer is that we want relations of the first two types involving unbounded vertices.  Similarly, the inclusion of the factor of is unnecessary as long as the generate this algebra, but it simplifies the writing of the second type of relations.

The category is, by definition, the category of finite-dimensional representations of .

By doing some clever combinatorics, one can prove that for Gale dual arrangements, these algebras are quadratic dual, and thus once one proves they are Koszul, they’ll be known to be Koszul dual.

Proving that these algebras are Koszul, and proving the derived equivalences is a slightly longer story, which I don’t think there’s time for that in a blog post.  You can read all about it in the paper.

Next time, I’ll try to show how these categories actually come from geometry.  While the paper linked above uses relatively little geometry, hypertoric varieties were in the forefront of our minds the whole time we were writing this paper, and they make it clear why anyone would ever have written down these algebras.

A few years ago, I gave a talk at NCSU on some work I had done on Littlewood-Richardson numbers, cluster algebras and such things. For the first half hour or so, I outlined the basic results I would be using about the representation theory of the group . Afterwards, I had a number of grad students thank me for this. So I’m going to try to turn that into a blog post (and enlarge it a little). The goal here is not to give you any proofs; rather, I want to get to the main results, show you how they connect and, above all, how to actually write down the representations of .

is, of course, the group of complex matrices with invertible determinant. We want to classify the linear representations of , meaning we want to find group homomorphisms from . Some examples: We have the trivial representation, where and every matrix in is mapped to the identity. We have the determinant representation, where again, and is mapped to . We have the standard representation, where and is mapped to itself. We have the dual of the standard representation, which is given in coordinates by . We have the symmetric representations, where acts on the symmetric powers of the standard representation, and the anti-symmetric or exterior representations, where acts on the anti-symmetric or wedge powers of the standard representation. If you have studied almost any field of math, I think it is safe to say that you have frequently encountered these examples; hopefully, that will suggest to you that classifying all representations of is a worthwhile problem.

Now, there is a technical point we have to get out of the way. If all we ask for is a group homomorphism, there are far too many. For example, we can use complex conjugation to get maps like . More confusingly, we can use other field automorphisms of , to get highly discontinuous maps. Another way things can get odd is that, as a group, has a lot of automorphisms (at least, if you believe in the axiom of choice), we could compose with any of these. Moreover, we could take any of these weird examples and tensor them with a normal example to get more weird ones. So we will want to come up with a rule that excludes these examples and limits us to the more algebraic examples of the preceding paragraph.

I’m an algebraic geometer, so my preferred fix is to require that the map is an algebraic map. This means that every entry of be given by a polynomial in the entries of and . (In most of the examples I gave of representations, the entries of are polynomials in the entries of , but in the dual representation we need to have as well.) One of the ways that I would defend my preferred choice is to point out that many seemingly different choices give the same result. You could also require that be holomorphic, and you would get the exact same set of maps. You could (this is the physicists’ choice) study the unitary group, , and require your maps to be continuous (or, alternately, smooth); then every representation you found would extend uniquely to an algebraic representation of . You could take the definition that I originally gave, using algebraic maps, and run it over any field of characteristic zero, and the description I will give in this post will still be correct. For that reason, I am trying to choose my notation to avoid mentioning the complex numbers whenever possible, although being perfectly consistent about this would be more of a pain than I think it is worth.

The first thing you need to know about is that it is what is called a reductive group. That means that any finite dimensional representation of splits as a direct sum of irreducible representations. (An irreducible representation is a representation which contains no subrepresentations other than and itself.) This splitting is unique in the appropriate sense, which takes a little effort to state correctly. For an example of a group that is not reductive, consider the group of complex numbers under addition; the representation can not be split into irreducible representations. So, we will be done if we can understand the irreducible representations of .

It isn’t just finite dimensional representations that are tamed by reductiveness. If is any algebraic variety (of finite type over ) and is an algebraic action of on , then it is easy to show that the coordinate ring, , of has an ascending filtration by finite dimensional representations. Reductiveness lets us split this filtration, so we get that is an infinite direct sum of finite dimensional irreducible representations. In particular, we can consider the action of on itself. Better, we can consider the action of on itself, with one copy of acting from the left and the other on the right. Explicitly, the action is .

(Those inverses are not where you expect them because the correspondence between and is contravariant. I’d advise you not to worry too hard about this.)

The Peter-Weyl theorem: the coordinate ring of , as a representation, is .

The sum runs over the isomorphism classes of irreducible representations of . I prefer to rewrite the summand as . (Note that the Wikipedia link, at least today, states this result in the analytic setting rather than the algebraic one. This is just another example of how which category you work in doesn’t matter very much for reductive groups.) If you have seen some representation theory, then you should be familiar with Peter-Weyl theorem in the setting of finite groups, where it states that the regular representation decomposes in this manner.

Let’s take an easy example. If , then is . The irreducible representations of are indexed by the integer ; the representation is .

There are two good ways to use the Peter-Weyl theorem to describe the representation of .

The first, analogous to the use of characters in the representation theory of finite groups, is to look at those functions on which are invariant under action of the diagonal of .

One the one hand, these are the functions such that . In other words, functions which depend only on the conjugacy class of their input. Now, the diagonalizable matrices are dense in , so any function on is determined by its values on the diagonalizable matrices. Furthermore, if is to be a conjugacy invariant function, then the value of on diagonalizable matrices is determined by its value on diagonal matrices. So we can describe such an by giving its value on . Finally, note that is conjugate to ; more generally, conjugation can reorder the entries of a diagonal matrix arbitrarily. So must be a symmetric function of the . Moreover, since we are working with algebraic maps and algebraic functions throughout, must be a symmetric Laurent polynomial in the . Conversely, any symmetric Laurent polynomial gives a conjugacy invariant polynomial function on .

On the other hand, from the presentation , we see that each irreducible representation contributes a single basis element to the diagonal invariants in . (Namely, the identity map from to itself.) Explicitly, is the trace of acting on . For example, the standard representation gives us and the dual of the standard representation gives us . The symmetric functions are called Schur functions; by the discussion of the previous paragraph, the Schur functions are a -basis for the symmetric Laurent polynomials.

Now, there is an obvious basis for the symmetric Laurent polynomials, namely, the monomial symmetric functions. There is one of these for each decreasing -tuple of integers, . We just take the sum where ranges over the permutations of .

So, on some intuitive level, we expect that there is one irreducible representation for each such -tuple . The idea which makes this precise is the idea of high weight vectors. If is a representation of then is called a high weight vector if is fixed by every upper triangular matrix with ’s on the diagonal.

Theorem 1: In every irreducible representation, up to scaling, there is a unique high weight vector.

For example, in the standard representation, is the high weight vector. It is also easy to check that the diagonal matrices send high weight vectors to themselves. So, if is an irreducible representation and its high weight vector, then for some .

Theorem 2: In the above setting, we always have . Conversely, for decreasing -tuple of integers, there is a unique irreducible representation such that the high weight vector transforms in this manner.

Now you know a set which is in bijection with the irreducible representations. But I promised I’d tell you how to write them down. The way to do this is the second good way to use the Peter-Weyl Theorem. We introduce the notation for the group of upper triangular matrices with ’s on the diagonal. So Theorem 1 tells us that, if we take invariants in , we get .

Now invariant elements of wind up corresponding to functions such that whenever . Some obvious functions with this property are the coordinate functions on the bottom row of our matrix: namely, , …, on . More subtly, any bottom justified minor is invariant under left multiplication by . For example, if and we write coordinates on as

,

then the determinant is left -invariant.

In fact, these determinants, along with , generate the ring of left -invariants in . There are many proofs of this, my favorite is Theorem 14.11 of Miller-Sturmfels.

So, the ring generated by these determinants is . How do you write down an individual ? You can extract this from everything I’ve said, but I’ll just tell you the answer. is the vector space spanned by products which use determinants of size , determinants of size and so forth, up to determinants of size . (Note that may be negative.)

(The statement that this recipe works is essentially the Borel-Weil Theorem in the algebraic category. More specifically, let be the group of upper triangular matrices. We showed that the space of functions on which transform in a certain way under the diagonal matrices is the vector space . Borel-Weil says that the sections of a certain line bundle on is the same vector space. The equivalence between these two viewpoints, compared to what came before, is not bad.)

Let’s wrap up with an example: we’ll take and . We must look at products which use one determinant of size one and one determinant of size two. Using the coordinates on above, we need to look at the vector space spanned by

, , , , , , , and .

These are 9 products here, but they span a vector space of dimension because

.

Choose 8 of these products to get a basis, and it is no trouble at all to write down . In particular, it is easy to compute the Schur function ; it is

.

As a final remark, there is no completely natural way to choose 8 of the 9 products above, and this problem only becomes worse as grows. There are some nice ways though, and one of the simplest is described in Corollary 14.9 of Miller-Sturmfels. An alternate way is the subject of my recent note with Kyle Petersen and Pavlo Pylyavskyy.

Since people who don’t get alerts from WordPress probably are no longer paying attention to this post, I thought I would top-post a comment left by Christine Cheng, a researcher who thinks about stable matchings. The full-text is below the fold. The main upshot is that there are some deterministic algorithms which produce fairer matchings than Gale-Shapley’s but they’re hard to implement.

I just discovered these exchanges a good five months from the last post. Hope my comments are still relevant and clear some things out.

Al Roth has done some work as to why stability is an important requirement to have for matchings. In the 1980’s, he compared some centralized matching algorithms in the U.K. for matching medical professionals with jobs, and found that those that did not produce stable matchings tend to have less and less participation over time while those that did continued to be in use. This make sense — afterall, the stability requirement was meant to prevent the unraveling of the matchings. What was surprising (and delightful) was that theory did predict practice. You can find relevant papers from his website and his book on stable matchings.

People have also looked into “fair” stable matchings. Greg kinda alluded to a natural one. Consider the set of all stable matchings. For each partipant a, let M(a) denote the partner of a in matching M. Let happiness(a, M) denote how happy a is with matching M. The egalitarian stable matching imaximizes the sum of happiness(a,M) over all a. The minimum regret stable matching maximizes the
minimum happiness(a,M) over all a (i.e., it maximizes the happiness of the unhappiest person). If happiness(a,M) is based on the rank of M(a), both problems are known to be solvable in polynomial time.

There is an intriguing one which I’ve worked on recently called the median stable matching. (See http://www.cs.uwm.edu/~ccheng) The basic idea is as follows: for each man m, rank his partners in all stable matchings from his most favorite to his least favorite. Note that he may have the same partner a in several stable matchings, in which case a will be present in this list multiple times as well. Now, let p_i(m) denote the ith woman in m’s sorted list. Let M_i consist of all pairs (m, p_i(m)). There’s no reason to believe that M_i is a matching — but it is! Furthermore, it is a stable matching! Additionally, let N denote the number of stable matchings of the instance. For simplicity, assume N is odd. Then M_{(N+1)/2} matches each man to his median stable partner, and it turns out, each woman to her median stable partner! BTW, this result is due to Teo and Sethuraman. Among all “fair” stable matchings that have been proposed, this was the most satisfying to me because the fairness occurs at a local level, as opposed to the previous ones where some person might still be better off than others.
Unfortunately, I show that finding the median stable matching is NP-hard. The problem is related to counting the number of order ideals of a poset, which is a #P-complete problem.

Finally, finding a random stable matching seems also a reasonable way to arrive at a fair stable matching. A recent paper by Bhatnagar et al (SODA 2008) shows that the MCMC method will take exponential time even when the preference lists of the participants are severely restricted.

One reader was curious if we had anything to say about the recent preprint by Xian-Jin Li entitled “A proof of the Riemann hypothesis”. Unfortunately, analytic number theory seems to be a weak spot of the mathematical blogosphere, so none of us seemed inclined to go through the paper and look for mistakes. Luckily, Terry Tao did and thinks he has found a mistake (which the author may claim to have fixed…things are starting to get a little confusing). Alain Connes also seems to be unconvinced. Oops.

Which leaves the rest of us to wonder what happened. I mean, this paper looked promising precisely because it didn’t look like the work of a crackpot. Li has a Ph.D. from Purdue (in mathematics) and is a mathematics professor at Brigham Young, and analytic number theory is his research area. He has several other unsuspicious articles on the arXiv, and the style of his Riemann hypothesis article is wholly unremarkable (considering that it claims to prove probably the most celebrated open problem still at large in the mathematical world). Why would someone risk the level of embarrassment involved in putting a proof of RH which had not been really thoroughly vetted on the arXiv, apparently with no warning (whether it can be fixed or not, if Terry Tao found a problem in less than 24 hours after it was placed on the arXiv, it definitely was not vetted thoroughly enough before being released on the world. It’s also on its 4th version on the arXiv in 3 days)? What was the hurry?

I can’t really speak to Li’s situation, since I don’t know the guy. It may well be that he sent his preprint to Tao and Connes and they didn’t get around to reading it. But if he didn’t, that was a huge mistake on his part, one which definitely makes him look more crackpotty than I expect he wants. If he didn’t give any conference talks on the subject before releasing the preprint, that was a huge mistake. Honestly, I think putting it on the arXiv, where it will remain forever, taunting him, rather than his personal webpage was something of a mistake. After all, you want a chance to get comments from the people who might be able to point out any mistakes you made before you end up on Slashdot. While this goes double, or perhaps n-uple for some large n if trying to prove an important problem like RH, I think it’s a good point in general that you should tell people about your work while it is still in its formative stages. It could save you a lot of pain. Admittedly, some people worry about being scooped, but I feel like this is the sort of thing that people are naturally more paranoid about than they should be. Ultimately, it would be better if we shared all our good ideas. After all, if somebody else does something cool with an idea you had, that just makes you look smarter for having such a good idea.

[Ed. - I changed the title of this post, since the original one was a bit more inflamatory than I intended]

Recently there have been some comments on our requests page:

Sander Kupers:

Maybe you could explain a bit about elliptic cohomology and topological modular forms…

and Thomas Riepe:

I would be curious about learning more on:
“… many constructions of classical algebra (eg, the theory of modular forms) are beginning to be seen to have deep homotopy-theoretic foundations.”…

Since this is somewhat related to some of my research, I have been recruited volunteered to talk about these things. The problem is that this is a Huge subject and a difficult subject and there is no way to adequately represent it in blog form. On the other hand it is a beautiful subject, filled with lots of exciting tidbits. Why not give it a go anyway, right?

I don’t know about you, but whenever I learn a big subject I find it much easier to understand if it is organized into sensible pieces. I’m going to try to do that on this blog, and the first sensible piece/topic is going to be generalized (co)-homology theories and why we should love them. Really I’m just going to ramble on about them and give some examples. The part about loving them will come later. In fact I won’t even talk about elliptic cohomology or TMF. These are examples of generalized cohomology theories which will come later. 

I’m going to assume that if you’re reading this, you’ve had a semester or two of algebraic topology (or are ambitious enough to teach yourself some of it). If you remember there are probably three things you studied/calculated: homotopy groups, homology, and cohomology. Each of these is a family of functors, one for each integer. The homology/cohomology functors and most of the homotopy functors take values in abelian groups (or the opposite category of abelian groups). A generalized homology theory is going to be the same kind of thing: a functor

But before talking about generalized homology let me ask you a question. Why do we like homology better then homotopy? Here’s a hint. What are the homology groups of ? There is a in dimension 2 and the rest are zero (I’m basically going to use reduced homology theories since it makes a few of the things I want to say easier). Okay, how about the homotopy groups of the 2-sphere? No body knows these.

Let’s do another example. Consider . This is a space where we know all the homotopy groups. It has a in dimension 2 and the rest is trivial. This is in some sense the dual of the last example. The 2-sphere has complicated homotopy and easy homology. A has complicated homology but easy homotopy. This is one manifestation of Eckmann-Hilton Duality.

The point is that while the homology of is more complicated then , it is not that complicated. Homology is computable. You probably had an exercise in your topology class where you compute the homology groups of this space, and with a few tricks it is not even that hard. Many people have it as a qual question (which means it doesn’t take, like, days to calculate).

So now we get to the real meat. Why? Why can we calculate with homology so easily but not homotopy?

The answer of course is that we have the Mayer-Vietoris axiom. Conceptually, what this property says is that if we know that a space is built from smaller pieces, then we can build the homology of the space from the homology of the pieces. Effectively this makes homology computable.

Let’s recall the axioms that the homology functor satisfies: A (reduced) homology functor consists of a functor, H, from the category of (nice) based spaces to the category of Abelian groups such that,

1. If is a (weak) homotopy equivalence then is an isomorphism.
2. If is a cofibration then,  is an exact sequence.
3. For each integer q, there is a natural isomorphism,.
4. If $X$ is a wedge of a set of (nice) based spaces, then the inclusions induce an isomorphism.
5. if and is zero otherwise.

This formulation can be found in May’s A Concise Course in Algebraic Topology. The last axiom is known as the dimension axiom. There is a uniqueness result for such functors. Any functor satisfying these axioms is isomorphic to ordinary homology. A generalized homology theory satisfies all the above axioms except we throw out the dimension axiom (5). That’s it. The exacness axiom (2) and suspension axiom (3) imply the Mayer-Vietoris exact sequence.

Now let’s look at some examples. One example you probably know is that we can replace the group with any abelian group G we like. You can even compute it exactly like homology, e.g. take the singular chain complex with G coefficients and then take homology. You have the same uniqueness result, again, but in more general examples of homology theories this uniqueness fails. A theory is not determined by its values on .

Here is a construction which I like very much. It is the above example but with coefficients in a topological Abelian group. Let A be a topological Abelian group, to a (pointed) space M we associate a new space of configurations of points in M labeled by A. Basically we take the spaces

which we think of as n distinct points of M, each with a label from A. Then we glue these spaces together so that, for example, we can h