Symmetric powers and the Chinese restaurant process

Here is a simple but cute computation that makes a good exercise in elementary representation theory with a probabilistic flavor — as usual, it’s probably well-known, but I wasn’t aware of it until doing the exercise myself.

We consider random permutations in the symmetric group S_n, and write \omega_n(\sigma) for the number of cycles in the disjoint cycle representation of \sigma\in S_n. It is a well-known fact (which may be due to Feller) that, viewed as a random variable on S_n (with uniform probability measure), there is a decomposition in law
where B_i is a Bernoulli random variable taking value 0 with probability 1-1/i and 1 with probability 1/i, and moreover the (B_i)‘s are independent.

This decomposition is (in the sources I know) usually proved using the “Chinese Restaurant Process” to describe random permutations (n guests numbered from 1 to n enter a restaurant with circular tables; they successively sit either at the next available space of one of the tables already occupied, or pick a new one, with the same probability; after all n guests are seated, reading the cycles on the occupied tables gives a uniformly distributed element of S_n.) If one is only interested in \sigma_n, then the decomposition above is equivalent to the formula
for the probability generating function of \omega_n. (This formula comes up in mod-poisson convergence of \sigma_n and in analogies with the Erdös-Kac Theorem, see this blog post).

Here is an algebraic proof of this generating function identity. First, z\mapsto \mathbf{E}(z^{\omega_n}) is a polynomial, so it is enough to prove the formula when z=m is a non-negative integer. We can do this as follows: let V be a vector space of dimension m over \mathbf{C}. Then define W=V^{\otimes n} and view it as a representation space of S_n where the symmetric group permutes the factors in the tensor product. Using the “obvious” basis of W, it is elementary that the character of this representation is the function g\mapsto m^{\sigma_n(g)} on S_n (the matrix representing the action of g is a permutation matrix). Hence, by character theory, \mathbf{E}(m^{\sigma_n}) is the dimension of the S_n-invariant subspace of W. Since the representation is semisimple, this is the dimension of the coinvariant space, which is the n-th symmetric power of V. This in turn is well-known (number of monomials of degree n in m variables), and we get
as claimed!

Update on a bijective challenge

A long time ago, I presented in a post a fun property of the family of curves given by the equations
where t is the parameter: if we consider the number (say N_p(t)) of solutions over a finite field with p\geq 3 elements, then we have
provided t is not in the set \{0, 4, -4\}. This was a fact that Fouvry, Michel and myself found out when working on our paper on algebraic twists of modular forms.

At the time, I knew one proof, based on computations with Magma: the curves above “are” elliptic curves, and Magma found an isogeny between the curves with parameters t and 16/t, which implies that they have the same number of points by elementary properties of elliptic curves over finite fields.

By now, however, I am aware of three other proofs:

  • The most elementary, which motivates this post, was just recently put on arXiv by T. Budzinski and G. Lucchini Arteche; it is based on the methods of Chevalley’s proof of Warning’s theorem: it computes N_p(t) modulo p, proving the desired identity modulo p, and then concludes using upper bounds for the number of solutions, and for its parity, to show that this is sufficient to have the equality as integers.  Interestingly, this proof was found with the help of high-school students participating in the Tournoi Français des Jeunes Mathématiciennes et Mathématiciens. This is a French mathematical contest for high-school students, created by D. Zmiaikou in 2009, which is devised to look much more like a “real” research experience than the Olympiads: groups of students work together with mentors on quite “open-ended” questions, where sometimes the answer is not clear (see for instance the 2016 problems here).
  • A proof using modular functions was found by D. Zywina, who sent it to me shortly after the first post (I have to look for it in my archives…)
  • Maybe the most elegant argument comes by applying a more general result of Deligne and Flicker on local systems of rank 2 on the projective line minus four points with unipotent tame ramification at the four missing points (the first cohomology of the curves provide such a local system, the missing points being 0, 4, -4, \infty). Deligne and Flicker prove (Section 7 of their paper, esp. Prop. 7.6), using a very cute game with products of matrices and invariant theory, that if \mathcal{V} is such a local system and \sigma is any automorphism of the projective line that permutes the four points, then \sigma^*\mathcal{V} is isomorphic to \mathcal{V}.

Not too bad a track record for such a simple-looking question… Whether there is a bijective proof remains open, however!


The Swiss Science Foundation (SNF in German, FNS in French) publishes a regular magazine that surveys current topics concerning science and research in Switzerland. It often highlights subjects that are unexpected and interesting in all areas of science, from the humanities to forestry, the deterioration of prussian blue in paintings and so on, and does so in its three parallel editions, in English, French and German.

The last issue has a special focus on Open Science in its various forms. For some reason, although there is no discussion of Polymath per se, the editors decided to have a picture of a mathematician working on Polymath as an illustration, and they asked me if they could make such a picture with me, and in fact two of them (the photographer is Valérie Chételat) appear in the magazine. Readers may find it amusing to identify which particular comment of the Polymath 8 blog I am feigning to be studying in those pictures…

Besides (and of greater import than) this, I recommend looking at the illustration pages 6 and 7,


which is a remarkably precise computer representation of the 44000 trees in a forest near Baden, each identified and color-coded according to its species… (This is done by the team of M. Schaepman at the University of Zürich).

Jordan blocks

Here is yet another definition in mathematics where it seems that conventions vary (almost like the orientation of titles on the spines of books): is a Jordan block an upper-triangular or lower-triangular matrix? In other words, which of the matrices

A_1=\begin{pmatrix}\alpha & 1\\0&\alpha\end{pmatrix},\quad\quad A_2=\begin{pmatrix} \alpha & 0\\1&\alpha\end{pmatrix}

is a Jordan block of size 2 with respect to the eigenvalue \alpha?

I have the vague impression that most elementary textbooks in Germany (I taught linear algebra last year…) use A_1, but for instance Bourbaki (Algèbre, chapitre VII, page 34, définition 3, in the French edition) uses A_2, and so does Lang’s “Algebra”. Is it then a cultural dichotomy (again, like spines of books)?

I have to admit that I tend towards A_2 myself, because I find it much easier to remember a good model for a Jordan block: over the field K, take the vector space V=K[X]/X^nK[X], and consider the linear map u\colon V\to V defined by u(P)=\alpha P+XP. Then the matrix of u with respect to the basis (1,X,\ldots,X^{n-1}) is the Jordan block in its lower-triangular incarnation. The point here (for me) is that passing from n to n+1 is nicely “inductive”: the formula for the linear map u is “independent” of n, and the bases for different n are also nicely meshed. (In other words, if one finds the Jordan normal form using the classification of modules over principal ideal domains, one is likely to prefer the lower-triangular version that “comes out” more naturally…)