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The conjugacy classes of the permutation group $S_n$ are indexed by partitions like $[6]$ and $[2,2,2] = [2^3]$ describing the cycle type. What happens when you take products of two whole conjugacy classes? I saw in a paper, $$[6][2^3] = 6[3,1^3] + 8[2^2,1^2]+5[5,1]+4[4,2]+3[3^2]$$ Which I take to mean if you multiply a 6-cycle (abcdef) and a product of disjoint 3-cycles (pq)(rs)(tv), you can get

  • a three-cycle (abc),
  • two two-cycles (ab)(cd),
  • a five-cycles (abcde),
  • a four-cycles and a two-cycle (abcd)(ef),
  • two three cycles (abc)(def)

With certain multiplicities. Is it predictable what kinds of conjugacy classes you get? Is there an interpretation of this as the intersection cohomology of some moduli space?

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I'm curious -- where did you see this? –  Michael Lugo Apr 18 '11 at 6:23
Those multiplicities are called connection numbers, and have been investigated by combinatorists. –  Mariano Suárez-Alvarez Apr 18 '11 at 6:45
You get some of them (the top ones, in some sense) as the structure coefficients of the cohomology algebra of the Hilbert scheme of points in the plane. Indeed, there is a filtration of the group algebra by the "age" of group elements, such that that cohomology algebra is the associated graded algebra to the induced filtration on the center of the group algebra. –  Mariano Suárez-Alvarez Apr 18 '11 at 6:50
Mariano: can you give some references for your comments? –  Amritanshu Prasad Apr 18 '11 at 6:58
Amritanshu -- I address this a bit at the end of my answer, but he's talking in part about: M. Lehn and C. Soerger. Symmetric groups and the cup product on the cohomology of Hilbert schemes. –  Paul Johnson Apr 18 '11 at 11:09
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3 Answers

up vote 17 down vote accepted

Short answer: Yes, on Hurwitz spaces.

Let's set these numbers up as the structure constants of $Z_d=Z(\mathbb{C}[S_d])$, the center of the group ring of the symmetric group $S_d$. The ring $Z_d$ has basis $K_{\mu}$, where $\mu$ is a partition of $d$, and $K_\mu$ represents the sum of all permutations of cycle type $\mu$. Then multiplication gives

$$K_\mu K_\nu=\sum_{\lambda} C^\lambda_{\mu,\nu} K_\lambda$$ for some numbers $ C^\lambda_{\mu,\nu}$, which are what you're interested in. I've seen these called connection coefficients, in work of Goulden and Jackson, for instance, their paper Transitive Factorisations into Transpositions and Holomorphic Mappings on the Sphere , which starts to get you a simple connection to geometry: by looking at ramified covers of the sphere. I'll talk about this a bit first, giving a rough sketch and some pointers, and then I'll address some of Mariano's comments.

This easiest connection to geometry is what you asked about in your previous question as "Hurwitz encoding", and David's answer there was good so I'll take that as background. You can start turning this into a problem about intersection theory by looking at Hurwitz Spaces.

You can make various flavours of these, but lets call the most basic one $H_{g,d}$, the moduli space of all holomorphic maps $\pi:\Sigma\to \mathbb{P}^1$ of degree $d$ from a smooth genis $g$ Riemann surface $\Sigma$ to the Riemann sphere $\mathbb{P}^1$. Generically, such maps will all have simple ramification, and by the Riemann-Hurwitz formula there will be $r=2g-2+2d$ such points of ramification, and so we see that $H_{g,d}$ will have complex dimension $r$. We will be able to view your numbers as suitable intersections on the Hurwitz space $H_{g,d}$.

There is a map from $H_{g,d}$ to $\mathbb{P}^r=(\mathbb{P}^1)^r/S_r$ that forgets $\Sigma$ and just remembers the $r$ branch points, (the critical values of $\pi$), counted with multiplicity. This is sometimes called the branch map, and I believe it is essentially what is known as the Lyashko-Looijenga map, and so I'll call this map LL.

The degree of the map LL is what is known as a Hurwitz number, and translating everything into monodromy we see that it counts the number of tuples of $r$ transpositions $t_i$ in $S_d$ with the product of the $t_i$ being the identity, divided by $d!$ coming from automorphisms of the cover, or choosing a labeling of the $d$ sheets of the cover, depending on your viewpoint.

To understand your connection coefficients geometrically, for a partition $\mu$ and a point $p\in \mathbb{P}^1$ we could define a cohomology class $\alpha(\mu, p)$ to consist of those maps in $H_{g,d}$ where $\pi$ has ramification profile $\mu$ over $p$. Then, if we've set up $g$ correctly with respect to $\mu, \nu$ and $\lambda$, the numbers $C^{\lambda}_{\mu,\nu}$ should be, again, up to some factor of automorphisms, the number of points in the triple intersection $\alpha(\mu,p_1)\cap \alpha(\nu,p_2)\cap \alpha(\lambda, p_3)$.

I'm not addressing some stuff (for isntance, connected versus discnonected covers) or necessarily giving you the most useful view in practice, but this is the simplest way to something like what you want, I think -- the Hurwitz space $H_{g,d}$ is not compact, and we'd want to compactify it (admissible covers is the first way, but this winds up not being normal, and you can use some orbifold Gromov-Witten theory and compactify with see twisted stable maps to get the normalization). But hopefully that's some idea of how this would go. To see this viewpoint used in practice, there are for instance, papers for instance of Lando and Zvonkine on the Arxiv -- I'm not sure where exactly you'd want to start. Through something known as the ELSV formula this story gets connected to intersection numbers on the moduli space of curves, which might be what you had in mind...

To connect into what Mariano was saying in comments, you'd want to get into the of a permutation $\sigma$ -- the minimal number of transpositions $\sigma$ factors into. Let's call this the weight of sigma -- for a permutation of cycle type $\mu$, it is equal to $|\mu|-\ell(\mu)$, where $\ell(\mu)$ is the number of parts of $\mu$. The center of the group ring $Z_d$ is filtered by the weight, and the "top" coefficients are ones where the weight adds -- where $d-\ell(\lambda)=d-\ell(\mu)+d-\ell(\nu)$.

In our geometric viewpoint, the weight is the amount of ramification above a certain point, and the top coefficients correspond to covers where all components are genus zero, the coefficients where the weight is off by two means we have a genus one cover, and similarly -- this filtration is geometrically filtering by the genus of our cover. The "top" coefficients are particularly nice in that they are independent of $d$ and so when you take the associated graded for each $s_d$ this plays well with the natural inclusions between the $S_d$ and you get some universal ring out of all the $S_d$ the Farahat-Higman ring.

Mariano's mention of the hilbert scheme of points in the plane is a bit of a different longer story here -- the brief outline as I like to think about it is that we can view $Z_d$ as the Chen=-Ruan orbifold cohomology of the $\mathcal{B}S_d=point/S_d$. The stack $\mathbb{C}^{d}/S_d$ will have the same vector space of cohomology, but the grading will be different -- this is what algebraic geometers call "age" that Mariano referred to. This age induces exactly the filtration above. The filtration is just doubled for $\mathbb{C}^{2d}$, and the Hilbert Scheme of points is a crepant resolution of this space, and so you get the relation on homology above. This is a long story, and seems slightly off from what you want.

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In the comments above you ask how to compute these connection numbers. Paul Johnson mentions that you can do this using $S_d$ character theory. I want to also point out his two papers Tropical Hurwitz Numbers and Chamber Structure of Double Hurwitz numbers, written together with Renzo Cavalieri and Hannah Markwig, are very nice and have a lot of explicit formulas.

They are answering a question that is a bit less general than you are asking. That question is the following: Let $\sigma$ be the partition $(2,1,1,\ldots,1)$ of $n$. Let $\lambda$ and $\nu$ be other partitions of $n$. What is the coefficient of $K_{\nu}$ in $K_{\lambda} K_{\sigma}^N$? This isn't quite the same as asking about $K_{\lambda} K_{\mu}$, but you would probably learn a lot from these papers anyways.

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Well, I had wanted the case that $\mu, \nu$ are the same. And I needed to sum over all conjugacy classes. It's possible, that since generic singuarities of curves will be double-points, all I need are products of transpositions. These papers will make interesting reading and I'll work with them. –  john mangual Apr 19 '11 at 15:27
Taking $\mu=\nu$ and then summing over all $\mu$ has connections to covers of a genus 1 curve, rather than a sphere -- essentially, you're degenerating a loop in the torus to a point, or cutting it along the loop, to get a sphere with two punctures. You dont' know what the original monodromy was around the loop you cut (summing over all of them), but to glue together you need them to be the same ($\mu=\nu$). Dijkgraaf connects with modular forms: citeseerx.ist.psu.edu/viewdoc/summary?doi= Mike Roth has a gentle intro mast.queensu.ca/~mikeroth/notes/covers.pdf –  Paul Johnson Apr 19 '11 at 16:33
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the number of the solution of the equations like abc = e in which a and b varies in two fixed conjugacy classes in a group G has a good description in terms of representations of the group G. So you are interested in the case G = S_n . A good reference for this is the book Galois cohomology by serre.

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