Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let be a partition. A semistandard Young tableau of shape is a filling of the Young diagram of with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau is defined as where is the total number of times appears in the tableau.

Definition 4:

where the sum is taken over all semistandard Young tableaux of shape .

As before we can readily verify that . This definition will allow us to deduce the Jacobi-Trudi identities for the Schur functions, which describe among other things the action of the fundamental involution . Since I’m trying to emphasize how many different ways there are to define the Schur functions, I’ll call these definitions instead of propositions.

Definition 5: .

Definition 6: .

The proof

The following proof is one of the standard applications of the Lindstrom-Gessel-Viennot lemma, and like much of the other material in these posts it can be found (with accompanying diagrams) in both Sagan and Stanley.

We’ll work with , the acyclic plane, with edges going northward and eastward only. We’ll also add some points which can be reached going northward from any finite point, and we won’t weight northward steps. But we will weight eastward steps as follows:

• In the e-weighting, each eastward step is weighted where is the sum of the coordinates of the destination vertex of the edge minus the sum of the coordinates of the initial vertex of the path. The weight of a path is the product of the weights of its eastward edges.
• In the h-weighting, each eastward step is weighted where is the difference between the -coordinate of the destination vertex of the edge and the -coordinate of the initial vertex of the path. Again the weight of a path is the product of the weight of its eastward edges.

Choose sources and sinks . Then every path from to determines a monomial in in the e-weighting and a monomial in in the h-weighting, and conversely every monomial arises in this way. So by the lemma it follows that the Jacobi-Trudi determinants count non-intersecting -paths .

Given such a non-intersecting -path we may construct two tableaux from it: one whose columns can be read off from the e-weighting of each path which has shape , and one whose rows can be read off from the h-weighting of each path which has shape . The non-intersecting condition is then precisely the condition that rows are weakly increasing and columns are strictly increasing; intersections would correspond to a violation of one or both of those constraints. (This is much easier to see with the corresponding diagrams; my apologies.) So in fact one obtains SSYTs, and moreover it’s not hard to show that all SSYTs of the correct shapes are obtained in this way.

The fundamental involution

Since the fundamental involution flips and , it follows that . There is a representation-theoretic interpretation of this. Recall that

and that

.

Applying the fundamental involution it follows that . We should interpret as the sign of an -cycle so that . Now the identity

gives

which has an obvious interpretation: the representation must come from by tensoring with the sign representation!

Relationship to a previous definition

We are now in a position to prove that Definition 4 is equivalent to one of the other definitions we gave earlier, namely Definition 3. We follow the proof given in Sagan, p. 164. First define to be with all the terms containing removed.

Lemma: Let be a composition and define the matrices . Then .

Proof. This is equivalent to the collection of identities

which is in turn equivalent to

which is obvious.

The equivalence of Definition 3 and Definition 4 is a simple corollary; one just picks so that is first and then , and then applies the first Jacobi-Trudi identity.

One of my favorite results in algebraic combinatorics is a surprisingly useful lemma which allows a combinatorial interpretation of the determinant of certain integer matrices. One of its more popular uses is to prove an equivalence between three other definitions of the Schur functions (none of which I have given yet), but I find its other applications equally endearing.

Let be a locally finite directed acyclic graph, i.e. it has a not necessarily finite vertex set with finitely many edges between each pair of vertices such that no collection of edges forms a cycle. For example, could be with edges and , which we’ll denote the acyclic plane. Assign a weight to each edge and assign to a path the product of the weights of its edges. Given two vertices let denote the sum of the weights of the paths from to . Hence even if there are infinitely many such paths this sum is well-defined formally, and if there are only finitely many paths between two vertices then setting each weight to gives a well-defined non-negative integer.

Let and be a collection of vertices called sources and vertices called sinks. We are interested in -tuples of paths, hereafter to be referred to as -paths, sending each source to a distinct sink. Let be the matrix such that . Then the permanent of counts the number of -paths, but this is not interesting as permanents are hard to compute.

A -path is called non-intersecting if none of the paths that make it up share a vertex; in particular, each is sent to distinct . A non-intersecting path determines a permutation of the vertices; let the sign of a non-intersecting -path be the sign of this permutation.

Lemma (Lindstrom, Gessel-Viennot): is the signed sum of the weights of all non-intersecting -paths.

Corollary: If the only possible permutation is (i.e. is non-permutable), then is the sum of the weights of all non-intersecting -paths.

Proof

The idea of the proof is straightforward: one defines a sign-reversing (weight-preserving) involution on -paths so that in the signed sum only the fixed points survive and the others cancel. (This is more or less equivalent to a use of inclusion-exclusion, but in many cases in combinatorics it is more natural to define the involution directly.) The involution is roughly as follows: if a path is non-intersecting, fix it (of course). Otherwise, there is some vertex that two paths share as a vertex. Flip the part of the paths after , which changes the sign of the induced permutation.

The only difficulty is to make sure this is really an involution, i.e. that the same portion of path gets flipped if one applies the map twice. To ensure this, let be the smallest index such that the path from intersects another path and let be the largest index of a path which intersects the above path. These indices remain unchanged after the involution, so the vertex also remains unchanged. If you want the full details, there is a nice introduction by Benjamin and Cameron in the AMM, Vol. 112, No. 6. Benjamin and Cameron describe a nice application to the evaluation of a determinant of Catalan numbers, so I won’t describe it; it makes more sense with the diagram anyway.

Applications

Gessel and Viennot’s original motivation was precisely an application to the acyclic plane, and indeed the lemma is sometimes stated for the plane only. Let and be strictly increasing sequences of non-negative integers and define sources and sinks . The number of paths from to is readily verified to be , so is the binomial determinant

Apparently these determinants have something to do with Chern classes. When either the or the are consecutive integers, this binomial determinant has a product formula (for example, see this AoPS thread), which Gessel and Viennot prove combinatorialy using their lemma.

The lemma (more or less) can be used to prove the matrix-tree theorem. I am told that this proof is in Aigner.

The lemma immediately implies that determinants are multiplicative. Actually, it implies an even stronger result, the Cauchy-Binet formula. Consider now three sets of vertices and construct two matrices , one for paths and the other for paths . Note that are not necessarily square, but is a square matrix counting paths which pass through some -vertices. Then counts nonintersecting paths which must therefore pass through distinct -vertices. If , there are no such paths. Otherwise, for every subset of of size , every -path factors as a product of -paths , and the signs multiply in the expected way. Letting denote the matrices counting such paths, it follows that

.

As a special case, if , then we obtain the identity

.

This identity implies that the square of the volume of a parallelpiped in with vertices given by the entries of is equal to the sum of the squares of the volumes of its projections to the standard copies of obtained by ignoring some of the coordinates. This is a high-dimensional generalization of the Pythagorean theorem, and it can also be used to prove the matrix-tree theorem. In fact, since the minors of a matrix to which Gessel-Viennot applies have almost exactly the same combinatorial interpretation as the full determinant, we even know that matrices of the form are positive-semidefinite. (More generally, this is true for any matrix coming from a nonpermutable graph.)

The lemma implies that non-intersecting random walks are a determinantal process, which connects them to many other mysterious processes. I wish I knew what to make of this.

Conjecture

Over the summer I conjectured that the lemma could be used to prove the identity

for the adjacency matrix of a finite graph . My idea was that one could construct a “blowup” of with the property that Gessel-Viennot applied, and Joel Lewis and I tried a few constructions that looked something like this: for each non-negative integer , has vertices where is a vertex of . For each edge of , has edges of weight . Finally, add the vertices and edges for each finite of weight .

Now it follows that a walk from to is precisely a walk of some length from to on of weight , hence is precisely equal to the entry of . So let the vertices be sources and the vertices be sinks.

Unfortunately, as constructed is not non-permutable. It’s possible one might be able to define a second sign-reversing involution on non-intersecting -paths in this setup, but I would really like it if could be modified to be nonpermutable. In all honesty I haven’t given this problem much thought since the summer, so it could be that the proof is relatively straightforward from here.

A few weeks ago on MathOverflow Greg Muller asked, “why do groups and abelian groups feel so different?” The answers were very interesting and came from several different perspectives, but I still don’t feel as if the question was resolved. So I’ll try to synthesize and summarize some of the answers and hopefully something will be clearer in the end.

Morphisms

Groups naturally arise as automorphism groups of objects in categories. From this perspective abelian groups should just correspond to objects with particularly simple symmetries. But abelian groups have their own special type of symmetry: the pointwise sum of two homomorphisms is also a homomorphism, so is enriched over itself, something that doesn’t hold at all for . What this implies is that the automorphisms of an abelian group have two composition structures: one coming from pointwise addition and another coming from composition. This is the “best” definition of a ring, at least in my opinion.

This observation also immediately tells us why is a special ring: it is the initial object in , since every abelian group has a -action given by . From here one is readily led to the “best” definition of a module, exactly analogous to how one might go from sets to permutations to groups to group actions. Now we can think of abelian groups as -modules and the action as a distinguished set of automorphisms of any abelian group. For some abelian groups these are the only automorphisms.

Non-abelian groups, on the other hand, have a totally different set of distinguished automorphisms: the inner automorphisms . Indeed, one way to define a non-abelian group is as a group with nontrivial inner automorphisms, and for some non-abelian groups these are the only automorphisms. In other words, while the structure of abelian groups is controlled in a strong way by the structure of (this is basically the content of the structure theorem), there is no such controlling object for general groups.

Perhaps precisely because their structure is so well-controlled, abelian groups lead to a lot of useful constructions in mathematics, such as abelian categories (the basic tool of homological algebra). Even to study groups one often passes to the category of modules over the group algebra!

What’s not clear to me is the following: should I think of abelianness as a powerful tool or as the easier version of a more difficult but more powerful theory of “non-abelian categories”?

A 2-categorical perspective

One particular complaint Greg had was that abelian groups rarely arise, in practice, as automorphism groups of objects. There is a way, however, one can get abelian groups “naturally” acting on things. The Eckmann-Hilton argument shows that two monoid structures on a set which are homomorphisms of each other must in fact be the same commutative monoid structure, and has the following consequences:

• The higher homotopy groups are abelian.
• The -morphisms in a -category with one object and one morphism form a commutative monoid. If the -morphisms are invertible, they form an abelian group.

The latter is due to the fact that -categories have two types of composition. This seems to have important philosophical implications; for example, awhile back at the n-Category cafe Bruce Bartlett asked a similar question about the difference between commutative and noncommutative algebras from an -categorical point of view. It seems as if to understand commutativity one should really move up the -categorical ladder; as Scott Carnahan puts it, if groups act on objects, then 2-groups act on categories, and the automorphisms of the identity functor automatically form an abelian group.

The big question, of course, is whether the abelian groups that naturally arise in mathematics can actually be fruitfully thought of in this way. Anyone have any thoughts?

The last time we talked about symmetric functions, I asked whether the vector space could be turned into an algebra, i.e. equipped with a nice product. It turns out that the induced representation allows us to construct such a product as follows:

Given representations of , their tensor product is a representation of the direct product in a natural way. Now, injects naturally into , which gives a new representation

.

The character of this representation is called the induction product of the characters of , and with this product becomes a graded commutative algebra. (Commutativity and associativity are fairly straightforward to verify.) It now remains to answer the first question: does there exist an algebra homomorphism ? And can we describe the inner product on coming from the inner product on ?

To answer this question we’ll introduce perhaps the most important class of symmetric functions, the Schur functions .

N.B. I’ll be skipping even more proofs than usual today, partly because they require the development of a lot of machinery I haven’t described and partly because I don’t understand them all that well. Again, good references are Sagan or EC2.

Schur-Weyl duality

One way that the Schur functions naturally occur is in the description of Schur-Weyl duality. To the extent that I understand this, it goes as follows. On there are two commuting representations: one of which acts on all the factors simultaneously and one of which permutes the factors. An important property of these representations is that they are mutual centralizers, i.e. each is the largest group of transformations which commutes with the other. It then turns out that the above representation decomposes as

where runs through all partitions with boxes and at most rows, is the irreducible representation of associated to the partition , and is a corresponding irreducible representation of .

We already know of two of these representations, the symmetric algebra and the exterior algebra . It turns out that the symmetric algebra corresponds to the partition (one row) and the exterior algebra corresponds to the partition (one column), and it’s not hard to see that the former is always there and the latter is only there as long as . As we saw previously, the characters of these representations can be described in terms of eigenvalues. This turns out to be true more generally.

Proposition: Suppose is a representation which is homogeneous, i.e. it respects scalar multiplication, and polynomial, i.e. every entry of the output is a polynomial function of the entries of the input . Then the character of is a symmetric function of the eigenvalues of .

The restriction to homogeneous polynomial representations is to avoid discontinuity and powers of the representation . Anyway, the above result is not too hard to see: since the diagonalizable matrices are dense and characters are both continuous and invariant under conjugation, the character must depend only on the multiset of eigenvalues, and since the representation is homogeneous and polynomial, the character must be as well.

I am not familiar with the explicit description of the representations , but the important point is that these representations are well-defined with respect to Schur-Weyl duality.

Definition 1: The Schur polynomial is the value of the character of evaluated at the diagonal matrix with entries .

For example, (the symmetric algebra) and (the exterior algebra). Note that if has more than rows then is not defined.

Of course, since I haven’t told you how to write these functions down, this is a rather unsatisfying definition, but at least it has a concrete tie to an important representation-theoretic concept. Now, it turns out that , so the Schur functions are well-defined as symmetric functions.

Schur-Weyl duality turns out to imply a strong relationship between these characters and the characters of the symmetric group which generalizes the relationship the complete homogeneous symmetric functions and elementary symmetric functions have to the trivial and sign representations.

Definition 2: Let denote the character of the representation of corresponding to evaluated at a permutation . Then

.

Determinantal formula

The Weyl character formulas imply the following determinantal formula for . First, some notation. Define

.

In particular is the Vandermonde determinant , and for any other partition the Vandermonde determinant divides it. (One could imagine these determinants appearing in the discussion of polynomial interpolation when certain coefficients are restricted to be zero, but I have never heard anyone talk about Schur functions this way.)

Definition 3: .

This definition has the advantage that it does not refer directly to representation theory, and it is also relatively straightforward to do computations with for small cases.

There are two other determinantal formulas for the Schur functions which are useful in studying, for example, the cohomology ring of a Grassmannian, but they can wait.

The characteristic map

We now define the characteristic map as follows: if is a class function on , then define

and extend by linearity. As we saw above, , so this is a natural definition from that perspective.

Proposition: . In other words, defines a ring homomorphism from with the induction product to .

The main technical detail of this proof is Frobenius reciprocity, but the point is that we have now found the relationship between and that we were looking for. With a little more work one can show that is bijective, and it follows that the form a basis of the symmetric functions.

The characteristic map takes the usual inner product on to the unique inner product on symmetric functions satisfying

.

The study of this inner product was initiated by Philip Hall and clarifies a number of results in symmetric function theory; see EC2, for example. But I haven’t digested this point enough to say anything meaningful.

I seem to have broken my MaBloWriMo streak. I hope you’ll believe me when I say it was impossible for me to get a post up yesterday. Unfortunately, the rest of this week looks just as hairy (for completely different reasons), so I’m going to have to take a break. Here’s where we’re headed once I have time again:

I’m going to skip a lot of background at some point and just introduce several equivalent definitions of the Schur functions. My hope is that stating some of the important results of the theory, even without proof, will be enough to get other people interested in symmetric function theory. I also get a lot of material to go back to and flesh out in subsequent posts, since I haven’t gone through most of the proofs of the basic results very thoroughly.

After that, I want to meander slowly through parts of basic algebraic number theory and algebraic geometry. My goal here is to thoroughly understand the classical analogy between rings of integers in number fields and nonsingular affine algebraic curves. Since several bloggers have covered much of this material in some form already, I’ll try to link to other posts I’m aware of, but I’ll have to repeat some things because I want to motivate every definition I need.

Since I don’t have anything else to say at the moment, let’s make this post an open thread and we’ll see if the Scott Aaronson style of blogging works for me. General comments, questions, suggestions, requests, etc. welcome!

Recall that the elementary symmetric functions generate the ring of symmetric functions as a module over any commutative ring . A corollary of this result, although I didn’t state it explicitly, is that the elementary symmetric functions are algebraically independent, hence any ring homomorphism from the symmetric functions is determined freely by the images of .

A particularly interesting choice of endomorphism occurs when we set . This endomorphism sends the generating function for the elementary symmetric polynomials to the generating function , and this is an involution – in other words, it follows that and itself must also be an involution on the ring of symmetric functions. Thus the elementary symmetric functions and complete homogeneous symmetric functions are dual in a very strong sense. This is closely related to the identity

and today I’d like to try to explain one interpretation of what’s going on that I learned from Todd Trimble, which is that “the exterior algebra is the symmetric algebra of a purely odd supervector space.”

Naive definitions

The multisets of a set with elements form a monoid under union, so we can take their monoid algebra, which is a generalization of the notion of group algebra. The homogeneous elements of degree are given by the set of linear combinations of multisets of size , and this gives the familiar construction of the symmetric algebra

where is the free vector space on a set with elements (and is the free commutative algebra on ). The Hilbert series of this algebra is ; in particular, the number of elements of degree is . So the symmetric algebra is a good linearization of the notion of multiset. One can think of it as the ring of polynomials , which is a familiar object.

On the other hand, if we try to linearize subsets in the same way we run into the problem of how to describe the product structure. The solution is to take a leaf out of the book of physics, namely the Pauli exclusion principle, to obtain a physically significant possibility: the product of two subsets with an element in common should be empty. In other words, the product should satisfy

for all . But it’s not hard to see that this implies that

.

In other words, the multiplication must be antisymmetric. This multiplication defines the exterior algebra on ,

.

The Hilbert series of this algebra is ; in particular, the number of elements of rank is . One way, therefore, to interpret the identity

is that the symmetric algebra on a vector space of dimension should behave something like the exterior algebra on a vector space of dimension , whatever that means. So what could that possibly mean?

Negative dimension

What we’re going to do is just start with a normal vector space which is “positive” and throw in a second vector space which is “negative.” This might be a sensible thing to do if, for example, you were interested in talking about a collection of positively and negatively charged particles. The supervector space we are looking at then is just the direct sum regarded as a -graded vector space; we’ll refer to the first component interchangeably as the even or positive part and the second component as the odd or negative part, and we’ll say it has dimension if , although what we’re really interested in is , which we’ll call the Euler characteristic and denote by .

Example. Let be a sequence of vector spaces and linear maps, and define the supervector space whose even part is all the even-numbered spaces and whose odd part is all the odd-numbered spaces. If the sequence is exact (with the zero vector spaces at the beginning and the end) then the Euler characteristic vanishes. This is the simplest manifestation of a deep relationship with homology theory; see the enlightening answers to another one of my MathOverflow questions here.

One way to think of how the category of supervector spaces should behave is to start with the category , since every representation decomposes into copies of the trivial representation (the even part) and the sign representation (the odd part). The intertwining operators here are the grade-preserving ones, i.e. the ones that send positive elements to positive elements and negative elements to negative elements; these are the morphisms in . (The Euler characteristic is then the trace of the non-identity element of .)

Next we should introduce a tensor product structure. Going off of how we want the -action to behave on each component of the tensor product, the even part of the tensor product should be

and the odd part should be

.

With this definition, Euler characteristics are multiplicative, so we have categorified the identity . Note that this is the same tensor product identity as the one we used to discuss the orthogonality relations. The unit of this tensor product is the base field regarded as a purely even space, so that , and one can check that all the obvious laws hold.

Supercommutativity

The next structure we’d like is a way to identify the tensor products and , i.e. a braiding. Unlike in the case for , where the only sensible identification sends to on pure tensors, there are two braidings. Let’s suppose for now that any sensible braiding will still have the form

for some scalar on pure homogeneous tensors. By bilinearity it really suffices to define how to switch two even elements, an even and an odd element, and two odd elements, since nice braidings satisfy . Now, compatibility with units (which are even) implies that

so it follows that ; in other words, even elements don’t introduce any sign changes. By the above considerations it follows that , hence for odd elements we have . The only way to avoid the boring braiding is to require that for odd elements, which is often written

where are the parities of (as elements of ). One can check that all the associativity axioms hold, so this braiding turns into a symmetric monoidal category, and now we can define the supersymmetric powers of a supervector space as the quotient of the tensor power by the above relation. Even better, we can now define the supercommutative algebra

.

Supercommutativity appears naturally in several settings: in the exterior algebra (which we already knew about), in the cup product in a cohomology ring, and in Clifford algebras, so there is plenty of evidence that this isn’t an ad hoc definition; see also the great answers when I asked why this definition was meaningful at MathOverflow.

Edit, 11/8/09: For a more concise and general way to describe , see also the answers to the MathOverflow question here.

And now something remarkable happens.

Proposition: .

If is purely even of dimension , then supercommutativity is commutativity and is a purely even copy of the usual symmetric power. If is purely odd of dimension , then supercommutativity is anticommutativity and is a purely -signed copy of the usual exterior power. This is the categorification of the set-multiset identity I wrote earlier, which we now know should really be written . So not only do we know how to interpret this result, but it has a great generalization, which we just have to prove.

Proof. In the general case where , any product of pure tensors can be rewritten, using supercommutativity, as the product of elements of followed by elements of , which gives a direct sum decomposition

where has Euler characteristic since the odd parts anticommute and the even parts commute. It remains to show that

.

The combinatorial proof for is as follows: we have a set of elements, of which are “bad” and the others are “good.” We want to compute the number of multisets with elements, all of which are good, and we proceed by inclusion-exclusion. We start with the multisets of arbitrary elements, then remove the multisets with at least one bad element, then add back the multisets with at least two distinct bad elements, and so forth. The proof for is identical except that we count subsets.

This proof should have a linearization in which we define morphisms making

an exact sequence (except at the ends), but I can’t figure out what the morphisms should be.

Characters

The upshot of all of this is that we can think of the symmetric and exterior algebras on an equal footing. There is a natural way to switch between the two since in the functor , where is the base field regarded as a purely odd vector space, interchanges the even and odd parts. In physics this is the relationship between the Fock spaces of fermions and bosons. As usual John Baez has lots to say on this subject, except that I can’t find the TWF I’m thinking of.

This is the same relationship as the one between the elementary symmetric functions and complete homogeneous symmetric functions. To see this, note that the symmetric and antisymmetric algebra constructions are functorial, so they send automorphisms to automorphisms and . It turns out that these are actually irreducible representations of and the traces of an element on these representations are the complete homogeneous symmetric functions and the elementary symmetric functions of the eigenvalues of , respectively.

Here’s a riddle for next time: the symmetric group acts on and by swapping factors, the first by the trivial representation and the second by the sign representation. These representations have characters the trivial and sign characters, and we showed way before that

and

.

In other words, the trace of the symmetric algebra has something to do with the trivial representation, and the trace of the exterior algebra has something to do with the sign representation. What does this mean?

I have a mathematical confession: I don’t trust uncountable sets.

Some time ago on MathOverflow somebody asked what a reasonable definition of “infinite permutation” would be. The first answer that comes to mind is a bijection . The set of all such bijections does form a group, but not only is it uncountably generated, it contains, as Darsh observes, a copy of every countably generated group (acting on itself by left multiplication). In particular it contains a copy of the free group on countably many generators. It also doesn’t seem to carry any natural kind of topology.

On the other hand, a much nicer candidate is the set of “compactly supported” permutations, i.e. those which fix all but finitely many elements. This countable group is generated by transpositions and therefore has a neat presentation given by the usual relations. I believe it’s also the largest locally finite subgroup of the full group of bijections.

I find this group much more philosophically appealing than the full group of bijections, and the reason is simple: each element of the group is computable. On the other hand, only countably many elements of the full group of bijections are computable: the rest can’t be written down by a Turing machine. And I don’t trust anything that can’t be written down by a Turing machine.

Corollary: I don’t trust the real numbers.

Instead of explaining what I mean by this, which I don’t think I have time for today, I’ll just throw a question out to the audience: how do you feel about all this?

In a previous post I gave essentially the following definition: given a discrete dynamical system, i.e. a space and a function , and under the assumption that has a finite number of fixed points for all , we define the dynamical zeta function to be the formal power series

.

What I didn’t do was motivate this definition, mostly because I hadn’t really worked out the motivation myself. Now that we have an important special case worked out, we can discuss the general case, which will give a purely combinatorial proof of the second half of the Newton-Girard identities.

The Euler product and the exponential formula

From the above definition we know that is an exponential generating function, i.e. the coefficient of is an integer. What we don’t know is whether this coefficient is always divisible by , i.e. whether is also an ordinary generating function. In fact, this is always true.

Because the zeta function ignores infinite orbits, we can suppose that every element of is in a finite orbit, i.e. for all there exists such that . The period of is defined to be the smallest such and will be denoted . The orbit of is the set and we will write for the size of an orbit.

Theorem (Euler product):

where the product runs over all orbits of . Equivalently, the coefficient of in counts the number of multisets of orbits of with total points.

As in the previous post, one can give a proof using Mobius inversion, but this is unenlightening. A more combinatorial approach is through the exponential formula, or at least through what we know about the cycle index polynomials of the symmetric groups. If we write , then we know that

where is the number of cycles of length in . In words, the coefficient of is the number of ways to assign to each cycle of length in a permutation of elements a sequence of points such that . This is because a point of period dividing uniquely determines its cycle.

It follows that any such assignment is uniquely determined by its points, which are a collection of not necessarily distinct points of each given a unique label from . Now here’s the important step: acts freely on labels, so in fact is an integer which counts the number of multisets of points of which are closed under the action of . We may group each point with its orbit (which will in general not be the same as its cycle), and then we have a multiset of orbits with total points as desired.

Note how nicely this generalizes the result from the previous post. If then a multiset of points closed under conjugation is precisely a monic polynomial with coefficients in , which is in turn precisely the generator of a maximal ideal of . More generally, if is a variety over then a multiset of points closed under conjugation is what is known as an effective divisor in algebraic geometry, and they form a free abelian monoid generated by the orbits. The Euler product is then the statement of “unique factorization” in this monoid.

Note also that this factorization is equivalent to the necklace congruences.

Complete homogeneous symmetric functions

The case we are interested in now is the case that is the set of aperiodic closed walks with a distinguished starting point on a finite graph with adjacency matrix and moves the starting point. Then counts the number of (not necessarily aperiodic) closed walks of length , hence . Note that these are the power symmetric functions of the eigenvalues of . Can we write down a similar expression describing the complete homogeneous symmetric functions in terms of combinatorial data?

In fact, we can. One way to define the complete homogeneous symmetric functions of the eigenvalues of a graph is as , the trace of the symmetric power of . It can be described combinatorially as follows: if has vertices , construct a new graph with vertices the set of all multisets of vertices of , where the edges between two multisets are precisely the -tuples of edges of sending one multiset to another. (This is precisely the description of the action of on the symmetric powers of the free vector space spanned by the vertices.) This gives the complete homogeneous symmetric functions as polynomial functions in terms of the entries of .

It then follows that is the number of closed walks of length on , i.e. the number of -tuples of edges that fix a multiset of vertices. But now grouping vertices by connecting the edges, we get a decomposition of the vertex set into a multiset of aperiodic closed walks. (That they are aperiodic is a consequence of the fact that different copies of the same vertex are treated as the same.) And these are precisely the multisets of points of closed under the action of . Combined with what we know about the zeta function, we have proven the following using only combinatorics.

Proposition: .

Taking derivatives now gives the second Newton-Girard identity, but we know from experience that this is equivalent to a pointing argument, which is as follows: in any multiset of aperiodic closed walks, fix a total order on the different copies of the same closed walk and point to a vertex . This vertex is contained in copies of some aperiodic closed walk , and it determines an arbitrary closed walk as follows: trace out the copies of starting from which are at least as large as the copy in which sits. Removing this closed walk gives an arbitrary multiset of aperiodic closed walks, and every multiset of aperiodic closed walks with a distinguished vertex arises uniquely in this way. It follows that

,

which is the second Newton-Girard identity. Note how this proof is nicely dual to the pointing proof of the first Newton-Girard identity.

The last step

The secret goal of this entire discussion was to push through a combinatorial proof of the identity

.

Unfortunately, I don’t quite see how to prove

with no further assumptions on , although it’s obvious by series manipulation. Here’s what I do know:

1. Foata gives a combinatorial proof in which (*) is done using the MacMahon Master Theorem. Unfortunately, I have yet to see a clear statement of this theorem, or at least one phrased in the language of walks on graphs.
2. On MathOverflow I asked a more general question about combinatorial interpretations of the symmetric functions of eigenvalues of a matrix. I was directed to the thesis of Andrius Kulikauskas, and this thesis agrees that (*) is very closely related to the MacMahon Master Theorem. Unfortunately, I have not yet had a chance to work through this paper, which is a shame since it’s exactly what I’ve been looking for; Lyndon words seem to be his primary tool as well.
3. One way to think about (*) is as the statement that in the determinant of , which counts all walks on , the cancellation of the terms is such that what’s left precisely describes multisets of aperiodic closed walks. Joel Lewis and I spent some time this summer trying to prove this using the Gessel-Viennot lemma, but to no avail. It’s possible a more general application of the involution principle or inclusion-exclusion might succeed.

In the style of the previous post, here’s a proof using tilings. Once more let describe tilings with tiles of length , so that is the companion matrix of . Then describes the number of tilings of length . We can think of tilings as words where letters can take up more than one position. With that philosophy, totally order the tiles. This defines the Lyndon words on tiles, and tilings then factor uniquely into a nondecreasing product of Lyndon words. But since here is the set of aperiodic tilings with a distinguished starting point, counts the multisets of Lyndon words of total length , and these two counts agree. It follows that

.

In particular, any tiling has a unique last tile, so we have shown that

which is precisely the identity in disguise. Once more we can now use the trick of lifting the to formal variables and setting them to be the elementary symmetric functions of another set of formal variables.

In number theory there is a certain philosophy that is a good toy model for the integers . The two rings share an important property: they are basically the canonical examples of Euclidean domains, hence PIDs, hence UFDs. However, many number-theoretic questions involving prime factorization over are much easier than their corresponding questions over . One way to see this is to look at their zeta functions.

The usual zeta function reflects the structure of prime factorization through its Euler product

where the product runs over all primes; this is essentially equivalent to unique factorization. Since we know that monic polynomials over also enjoy unique factorization, it’s natural to ask whether there’s a sum over monic polynomials that would give a similar Euler product.

In fact, there is such a product, and investigating it leads naturally to a seemingly unrelated subject: the combinatorics of words.

The cyclotomic identity

To phrase the Euler product correctly, recall that the zeta function of a ring of integers in a finite extension of can be written

.

where the sum runs over all ideals of , the product runs over all prime ideals of , and is the norm of an ideal; this Euler product reflects prime factorization of ideals in , and to make it work it is crucial that the definition of norm be multiplicative. On the other hand, there is an obvious choice of multiplicative function on polynomials: if the degree of the polynomial is , define its norm to be . (The choice of sign will make sense later.)

Since every ideal in is principal, we may identify ideals with monic polynomials. There are monic polynomials of degree , so we are led to the zeta function

.

This is almost disappointing: it’s much easier to deal with than the Riemann zeta function. Analytic continuation and the functional equation are trivial, and we know all of its poles and zeroes! But let’s keep going. The Euler product of the above zeta function is given by

where is the number of irreducible polynomials over of degree . I claim that knowing only that this product is equal to uniquely determines for all . In fact, much more is true.

Proposition: The group of formal power series of the form is isomorphic to the direct product of countable copies of with generators .

Proof. Let be the group of formal power series of the form . It’s straightforward to verify that where the element is sent to (this is an internal direct product), and the conclusion follows from here. More generally we can replace with any power series of the form .

We can now compute as follows. Taking times the logarithmic derivative of the zeta function gives

and the coefficient of on both sides is . How should we interpret this identity? Here’s one way: the polynomial factors over as the product of all irreducible polynomials of degree . In other words, every element of has a unique period under the action of the Frobenius map, and irreducible polynomials are in natural bijection with orbits of the Frobenius map. This is a very important point to which I will return. Mobius inversion then gives

.

Hence we have proven the cyclotomic identity

and our explicit formula for gives, rather easily, the asymptotic

where , by analogy with the usual prime number theorem.

Lyndon words

revealed itself to be Moreau’s necklace-counting function, which also counts equivalence classes of aperiodic necklaces. I was curious if there was an explicit bijection here and asked MathOverflow. It turns out that the bijection has some interesting things to say about finite fields, as I described in a different question.

The key is the Frobenius map . One can think of the algebraic closure , which is the variety associated to the ring , as a dynamical system on which the Frobenius map acts, and this dynamical system has the property that the fixed points of are precisely a copy of ; this is a fundamental fact of Galois theory and implies that the dynamical zeta function (which we first encountered when talking about Newton’s sums) is equal to

.

In fact, this definition agrees with the definition using ideals. The connection, as I have said, is that irreducible polynomials can naturally be identified with orbits under the action of the Frobenius map. (This is a simple version of the ideal-variety correspondence over non-algebraically closed fields: maximal ideals of can naturally be identified with “closed points” in .)

For aperiodic necklaces, the relevant dynamical system is given instead by the set of aperiodic words over an alphabet of size , which are words of length which have no nontrivial stabilizer under cyclic rotation. These words are acted on by cyclic rotation , and the fixed points of can be identified with aperiodic words of period , which can in turn be identified with words of length (since any word has a unique decomposition into copies of an aperiodic word), of which there are . It follows that, again,

.

The connection between cyclic rotation and the Frobenius map is as follows. Fix an irreducible monic polynomial of degree . If one of its roots is , then the rest of its roots are , and has basis , so we can write any of its elements as

.

In this basis, the Frobenius map acts by cyclic rotation of the coordinates , which define a word of length on an alphabet of size , and the number of elements in the orbit is precisely the length of the unique aperiodic factor of this word. In particular, orbits of length are in bijection with the other irreducible polynomials of degree , which in turn are in bijection with equivalence classes of aperiodic words of length .

A concrete way to think about equivalence classes of aperiodic words is as follows. Fix a total ordering on the alphabet, and consider the lexicographical order on words, which is total. Of the cyclic rotations of an aperiodic word, a unique word is minimal in the lexicographic order, and such words are referred to as Lyndon words. Now, what all this zeta function stuff suggests is that, since the coefficient of is equal to , words should have a unique factorization of some sorts into Lyndon words with multiplicity.

This is in fact true. The more precise statement is as follows.

Proposition: Fix a totally ordered alphabet . Every word on has a unique factorization where is a Lyndon word and in the lexicographic order for all .

Proof. We proceed by induction. Define to be the minimal word such that . We claim that is a Lyndon word. is aperiodic since for any words . Moreover, if is not Lyndon then there would exist such that , and then , so is not minimal; contradiction.

By induction we obtain a factorization into Lyndon words such that for all , so we have shown existence. To show uniqueness, suppose that is some other factorization, and suppose WLOG that . The only way this could occur is if is a left factor of , since otherwise their letters would disagree, but this is impossible by the assumption that is minimal.

It therefore follows that a word is uniquely specified by the multiplicity with which each Lyndon word appears in it, and this statement is equivalent to the cyclotomic identity.

As we saw last time, a sequence of nested groups gives rise to a graded “poset” in which objects can be related by more than one arrow. The poset we want to construct now is given by the sequence of symmetric groups (where and are both the trivial group). Since irreducible representations of the symmetric groups are parameterized by Young diagrams, this gives the set of Young diagrams a graded structure where the elements of rank are the Young diagrams with boxes.

It turns out that this “poset” is a genuine poset; in other words, in the restriction of an irreducible representation of to an irreducible representation of , the resulting representation contains each irreducible representation at most once. In fact, much more is true. In a poset we write if and there is no with ; this relation is called covering. So elements of rank cover elements of rank .

Theorem (Branching Rule): if and only if is obtained from by removing a box.

This is the remarkable theorem that really justifies the choice of Young diagrams as a natural description of the irreducible representations of the symmetric group. Today we’ll explore some of the consequences of this theorem.

Let denote the rank of order , where contains only the empty partition . Let denote the free vector space on , which is graded by rank, hence

.

We can think of elements of as generalizing direct sums of irreducible representations of . From this perspective, comes equipped with two distinguished linear operators such that

and

.

restrict to operators and and are essentially the induction and restriction maps. Of course, these maps (which one can refer to as “up” and “down,” respectively) can be defined for any graded poset, but part of the reason we like Young’s lattice is that its up and down operators carry special significance. Frobenius reciprocity is visible in the statement that .

The differential property

The operators have a surprising property whose significance was first recognized by Stanley in 1988 (!).

Proposition: .

Proof. Suppose a Young diagram has boxes on its southeast boundary. If we add a box and then remove a second box, we could also have removed the second box and added the first box, so these terms in cancel. If we add and remove the same box, there are ways to do this. However, if we remove and add the same box, there are ways to do this. Hence as desired.

Note that this is essentially the same balls-in-bags argument as in the previous post and establishes the following remarkable fact.

Fact: The action of on is a representation of the Weyl algebra.

I asked what this means over at MathOverflow, but Ben Webster’s reply was a little over my head. (If somebody wants to add their own answer, that would be great!) Posets with this property are called -differential because behaves like ; see Stanley’s paper for more discussion.

The upshot of all of this is that polynomials in can be written as a unique linear combination of terms of the form by repeated application of the above relation. This is immensely useful for counting walks on Young’s lattice, which correspond to interesting combinatorial information about tableaux. For example,

because a Young tableaux of shape is precisely a sequence of ways to build the corresponding Young diagram by adding boxes. (Thus is the multiplicity of the irreducible representation corresponding to in the induced representation of from the trivial representation of , which is the regular representation.)

More generally, given a word on , a Hasse walk of type is a walk on Young’s lattice starting at the empty partition in which one takes a step of type (add or remove a box), then a step of type (add or remove a box), and so forth. Clearly not all words give rise to valid walks; those words that do will be called valid, and it’s not hard to characterize them. We let denote the number of such walks which end at , hence where ranges over all partitions of the right size.

Theorem: Given a valid word , let . For each , let be the number of s to the right of and let be the number of s to the right of . Then

.

Corollary: .

The proof can be found in Stanley’s notes, so I won’t repeat it except to say that it depends only on the differential property. Moreover, it does not depend at any point on the fact that the symmetric groups have size . In the notes and in his paper, Stanley describes a number of other combinatorial results one can arrive at in this way, but they would take us too far afield.