Page 36 – E. Kowalski's blog

The Legendre polynomials are ubiquitous

One can define the $j$ -th Legendre polynomial $P_j$ in many ways, one of the easiest being to use the generating function
$\sum_{j\geq 0}{ P_j(x)T^j}=(1-2xT+T^2)^{-1/2}.$
Like many “classical” special functions (which one might call “the functions in Whittaker & Watson” — I find it charming, incidentally, that the PDF of this edition has exactly 628 pages), these can also be defined using representation theory. This is done by considering the group $G=\mathrm{SU}_2(\mathbf{C})$ and its $(2j+1)$ -dimensional irreducible representation on the space $V_{2j}$ of homogeneous polynomials in two variables of degree $2j$ , where the action of $G$ is by linear change of variable:
$(g\cdot P)(X,Y)=P((X,Y)g)=P(aX+cY,bX+dY)$
for
$g=\begin{pmatrix}a&b\\c&d\end{pmatrix}.$
Then, up to normalizing factors, $P_j$ “is” the matrix coefficient of $V_{2j}$ for the vector $e=X^jY^j\in V_{2j}$ . Or, to be precise (since a matrix coefficient is a function on $G$ , which is 3-dimensional, while $P_j$ is a function of a single variable), we have
$P_j(\cos\theta)=c_j \langle g_{\theta}\cdot e,e\rangle$
for the elements
$g_{\theta}=\begin{pmatrix}\cos(\theta/2)&i\sin(\theta/2)\\i\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}$
(the inner product $\langle \cdot,\cdot\rangle$ used to compute the matrix coefficient is the $G$ -invariant one on $V_{2j}$ ; since this is an irreducible representation, it is unique up to a non-zero scalar; the normalizing constant $c_j$ involves this as well as the normalization of Legendre polynomials.) For full details, a good reference is the book of Vilenkin on special functions and representation theory, specifically, Chapter 3.)

Note also that, since $e$ is, up to a scalar, the only vector in $V_{2j}$ invariant under the action of the subgroup $T$ of diagonal matrices, one can also say that $P_j$ is “the” spherical function for $V_{2j}$ (with respect to the subgroup $T$ ).

This seems to be the most natural way of recovering the Legendre polynomials from representation theory. Just a few days ago, while continuing work on the lecture notes for my class on the topic (the class itself is finished, but I got behind in the notes, and I am now trying to catch up…), I stumbled on a different formula which doesn’t seem to be mentioned by Vilenkin. It is still related to $V_{2j}$ , but now seen as a representation of the larger group $\mathbf{G}=\mathrm{SL}_2(\mathbf{C})$ (the action being given with the same linear change of variable): we have

$P_j(1+2t)=d_j \langle u_t\cdot e,u_1\cdot e\rangle$

where $d_j$ is some other normalizing constant, and now $u_t$ are unipotent elements given by
$u_t=\begin{pmatrix}1&t\\ 0 & 1\end{pmatrix}.$

It’s not quite clear to me where this really comes from, though I suspect there is a good explanation. Searching around the web and Mathscinet did not lead, in any obvious way, to earlier sightings of this formula, but it is easy enough to get thoroughly unenlightening proof: just use the fact that
$u_t\cdot e=X^j(tX+Y)^j,$
expand into binomial coefficients, use the formula
$\langle X^iY^{2j-i}, X^kY^{2j-k}\rangle=\binom{2j}{i}^{-1}\delta(i,k),$
for the invariant inner-product, and obtain a somewhat unwieldy polynomial which can be recognized as a multiple of the hypergeometric polynomial
${}_2F_1(-j,1+j;1;-t),$
which is known to be equal to $P_j(1+2t).$ (Obviously, chances of a computational misake are non-zero; I certainly made some while trying to figure this out, and stopped computing only when I got this nice interesting result…)

Birds of the Léman

My last post about birds concerned the great crested grebe (which appears in a most hilarious way in Scoop — “Lord Copper,” he was saying, “no man shall call me a liar unchastised. The great crested grebe does hibernate.”)

Although a more recent week-end excursion on the Léman was richer in raptors, it was also a pleasure to watch the noble heron (Héron cendré, I guess) guarding the piers:

However, the thrilling part was to watch raptors fishing; to be precise, I think most of them were the same species

which — interpreting my bird-book — is quite likely to be the Milan noir (Milvus migrans) (it might be the Milan royal, but that seems much more unlikely). Of course, most of my attempts to photograph the fishing of the milans resulted in blurry pictures of the surface of the lake; however, some succeeded, like this one

or that one

Subtile finesse de la notation…

For a long time (since I first heard of it around 1990), I thought that the terminology “Property (T)” was a completely arbitrary name, and no better than the thousand notions of “admissible thingummy” or “good khraboute” which sprinkle too many mathematics papers. Then I learnt a few years ago that the “T” was supposed to refer to the Ttrivial representation, since — in a suitable language — the property is about the trivial representation of a group $G$ .

This was already better. But much more recently, I learnt from S. Mozes that the typography “(T)” itself was not some random choice, but was meant to express the fact that the trivial representation $T$ is supposed to be alone in some open set, incarnated as an open interval (so one should read $()$ as in $(a,b)$ )…

A direct corollary is that the right translation in French is, of course, Property $]T[$ .

By the way, I personally much prefer the $]a,b[$ , $[a,b]$ , $]a,b]$ , etc, notation for intervals, but I’m told that many find this ugly beyond belief and much prefer the $(a,b)$ , $(a,b]$ , etc, style…

Multiplicativity, where art thou?

The inequality I was pondering during my recent vacations is another result of Burnside: for a finite cyclic group $G=\mathbf{Z}/m\mathbf{Z}$ , denoting by $S$ the set of generators of $G$ , we have
$\sum_{x\in S}|\chi_{\rho}(x)|^2\geq |S|=\varphi(m)$
for any finite-dimensional representation $\rho$ of $G$ with character $\chi_{\rho}$ , provided the latter does not vanish identically on $S$ (this may happen, e.g., for the regular representation of $G$ ). This was used by Burnside (and still appears in many books) in order to prove that if an irreducible representation $\rho$ of a finite group has degree at least $2$ , its character must be zero on some elements of the group. (The cyclic groups $G$ used are those generated by non-trivial elements of the group, and the representations are the restrictions of $\rho$ to those cyclic groups; thus it is an interesting application of reducible representations of abelian groups to irreducible representations of non-abelian ones… the lion and the mouse come to mind.)

This inequality can not be considered hard: it is dispatched in two lines invoking Galois theory to argue that
$\prod_{x\in S}{|\chi_{\rho}(x)|^2}$
is a (non-negative) integer, and the arithmetic-geometric inequality to deduce that
$\frac{1}{|S|}\sum_{x\in S}{|\chi_{\rho}(x)|^2}\geq \Bigl(\prod_{x\in S}{|\chi_{\rho}(x)|^2}\Bigr)^{1/|S|}\geq 1$
unless this product is zero.

However, if we view this as a property of representations of a finite cyclic group, one (or, at least, I) can’t help wonder whether there is a “direct” proof, which only involves the formal properties of these representations, and does not refer to Galois theory. This is what I was trying to understand while basking in the italian spring and gardens. As it turns out, there is is in fact quite a beautiful argument, which is undoubtedly too complicated to seriously replace the Galois-theoretic one, but is directly related to very interesting facts which I didn’t know before — I therefore consider well-spent the time spent thinking about this inequalitet. (Philological query: what is the diminutive of “inequality”? or of “formula”?)

The idea is that a representation $\rho$ of $G$ is a direct sum
$\rho=\bigoplus_{a}{n(a)\chi_a},$
where $a$ runs over $\mathbf{Z}/m\mathbf{Z}$ , and $\chi_a$ is the character
$x\mapsto e(ax/m),$
for some integral multiplicities $n(a)$ . Hence the left-hand side of Burnside’s inequality, namely
$\sum_{x\in S}{|\chi_{\rho}(x)|^2}$
can be seen as a quadratic form in $m$ integral variables, and the question we ask is: what is the minimal non-zero value it can take? (Note that the example of the regular representation shows that this quadratic form, though obviously non-negative, is not definite positive.)

Now the funny part of the story is that this quadratic form, say $Q_m$ , is naturally a tensor product
$Q_m=\bigotimes_{p^{k_p}\mid\mid m}{Q_{p^{k_p}}}$
of the corresponding quadratic forms arising from the primary factors of $m$ . Hence we have a problem about a multiplicatively defined quadratic form, and the answer (given by Burnside) is rhe Euler function $\varphi(m)$ , which is also multiplicative; should it not then feel natural to treat the case of $m=p^k$ for some prime $p$ , and claim that the general case follows by multiplicativity?

Alas, a second’s thought shows that it is by no means clear that the minimum of a tensor product of (non-negative, integral) quadratic forms should be the product of the minima of the factors! Denoting by $s(Q)$ the minimum of $Q$ (among non-zero values), we certainly have
$s(Q_1\otimes S_2)\leq s(Q_1)s(Q_2),$
but the converse inequality should immediately feel doubful when the forms are not diagonalized. And indeed, in general, this is not true! I found examples in the book of Milnor and Husemoller on bilinear forms (which I had not looked at for a long time, and I am very happy to have been motivated to look at it again!); they are attributed to Steinberg, through a result of Conway and Thompson, itself based on the famous Siegel formula for representation numbers of quadratic forms…

However, despite this fact about general forms, it turns out that the quadratic forms $Q_{p^k}$ have the property that
$s(Q_{p^k}\otimes Q')=s(Q_{p^k})s(Q')$
for all other quadratic forms $Q'$ (always non-negative integral). This I found explained by Kitaoka. Here the nice thing is that it depends on writing $Q_p$ as (essentially) a variance
$Q_p(n)=p\sum_{a}{n(a)^2}-\Bigl(\sum_{b}{n(b)}\Bigr)^2$
of the coefficients $n(a)$ , and using the alternate formula
$Q_p(n)=\frac{1}{2}\sum_{a,b}{(n(a)-n(b))^2}\quad\quad\quad\quad\quad\quad (\star)$
which corresponds, in probability, to the formula
$\mathbf{V}(X)=\frac{1}{2}\mathbf{E}((X-Y)^2)$
for the variance of a random variable $X$ , using an independent “copy” $Y$ of $X$ (i.e., $Y$ is has the same probability distribution as $X$ , but is independent.) This probabilistic formula, I have to admit, I didn’t know — or had forgotten –, though it is very useful here! Indeed, the reader may try to prove directly that $Q_p(n)\geq p-1$ unless it is $0$ , using any of these formulas for $Q_p(n)$ ; I succeeded with the other formula
$Q_p(n)=p\sum_{a}{\Bigl(n(a)-\frac{1}{p}\sum_{b}{n(b)}\Bigr)^2}$
but the argument was much uglier than the one with $(\star)$ above! (It is also clearer from the latter that $Q_p$ takes integral values for integral arguments, despite the $1/p$ which crop up here and there…)

For more details, I’ve written this up in a short note that I just added to the relevant page.

Lago Maggiore

Mathematicians coming to Verbania, on the Lago Maggiore, for tourism may be interested either in Riemann’s grave, in borromean rings or in getting some rest. Charmingly, these can all be combined at will. I haven’t yet visited the first, but visiting the Borromean islands

naturally leads to

encounters

with the

second. As for resting, I had an amusing elementary inequality in mind (almost) wherever I went, and that was actually rather nice…

Another highlight was the opportunity to witness the mating display of the Great Crested Grebe (or grèbe huppé, or Podiceps cristatus)

something which — according to my bird book — is not so common. I even took a short movie, but in true Murphy’s Law fashion (probably to compensate for its failure when photographing geckos making hand signals) I just missed the few seconds when the birds actually stood on the lake (as in this picture, though the ones I saw did not hold a fish at that time).

A restaurant recommendation in Verbania-Intra is the Trattoria Concordia; bear in mind that local red wines are, typically, slightly fizzy; it can make for a real vacation feel…