Tag: representations

THE rationality problem

Published April 27, 2007 by lieven

This morning, Esther Beneish
arxived the paper The center of the generic algebra of degree p that may contain the most
significant advance in my favourite problem for over 15 years! In it she
claims to prove that the center of the generic division algebra of
degree p is stably rational for all prime values p. Let me begin by
briefly explaining what the problem is all about. Consider one n by n
matrix A which is sufficiently general, then it will have all its
eigenvalues distinct, but then it is via the Jordan normal form theorem uniquely
determined upto conjugation (that is, base change) by its
characteristic polynomial. In
other words, the conjugacy class of a sufficiently general n by n matrix
depends freely on the coefficients of the characteristic polynomial
(which are the n elementary symmetric functions in the eigenvalues of
the matrix). Now what about couples of n by n matrices (A,B) under
simultaneous conjugation (that is all couples of the form $~(g A
g^{-1}, g B g^{-1}) $ for some invertible n by n matrix g) ??? So,
does there exist a sort of Jordan normal form for couples of n by n
matrices which are sufficiently general? That is, are there a set of
invariants for such couples which determine it is freely upto
simultaneous conjugation?

For couples of 2 by 2 matrices, Claudio Procesi rediscovered an old
result due to James Sylvester saying
that this is indeed the case and that the set of invariants consists of
the five invariants Tr(A),Tr(B),Det(A),Det(B) and Tr(AB). Now, Claudio
did a lot more in his paper. He showed that if you could prove this for
couples of matrices, you can also do it for triples, quadruples even any
k-tuples of n by n matrices under simultaneous conjugation. He also
related this problem to the center of the generic division algebra of
degree n (which was introduced earlier by Shimshon Amitsur in a rather
cryptic manner and for a while he simply refused to believe Claudio’s
description of this division algebra as the one generated by two
_generic_ n by n matrices, that is matrices filled with independent
variables). Claudio also gave the description of the center of this
algebra as a field of lattice-invariants (over the symmetric group S(n)
) which was crucial in subsequent investigations. If you are interested
in the history of this problem, its connections with Brauer group
problems and invariant theory and a short description of the tricks used
in proving the results I’ll mention below, you might have a look at the
talk Centers of Generic Division Algebras, the rationality problem 1965-1990
I gave in Chicago in 1990.

The case of couples of 3 by 3 matrices was finally
settled in 1979 by Ed Formanek and a
year later he was able to solve also the case of couples of 4 by 4
matrices in a fabulous paper. In it, he used solvability of S(4) in an
essential way thereby hinting at the possibility that the problem might
no longer have an affirmative answer for larger values of n. When I read
his 4×4 paper I believed that someone able to prove such a result must
have an awesome insight in the inner workings of matrices and decided to
dedicate myself to this problem the moment I would get a permanent
job… . But even then it is a reckless thing to do. Spending all of
your time to such a difficult problem can be frustrating as there is no
guarantee you’ll ever write a paper. Sure, you can find translations of
the problem and as all good problems it will have connections with other
subjects such as moduli spaces of vectorbundles and of quiver
representations, but to do the ‘next number’ is another matter.

Fortunately, early 1990, together with
Christine Bessenrodt we were
able to do the next two ‘prime cases’ : couples of 5 by 5 and couples of
7 by 7 matrices (Katsylo and Aidan Schofield had already proved that if
you could do it for couples of k by k and l by l matrices and if k and l
were coprime then you could also do it for couples of kl by kl matrices,
so the n=6 case was already done). Or did we? Well not quite, our
methods only allowed us to prove that the center is stably rational
that is, it becomes rational by freely adjoining extra variables. There
are examples known of stably rational fields which are NOT rational, but
I guess most experts believe that in the case of matrix-invariants
stable rationality will imply rationality. After this paper both
Christine and myself decided to do other things as we believed we had
reached the limits of what the lattice-method could do and we thought a
new idea was required to go further. If today’s paper by Esther turns
out to be correct, we were wrong. The next couple of days/weeks I’ll
have a go at her paper but as my lattice-tricks are pretty rusty this
may take longer than expected. Still, I see that in a couple of weeks
there will be a meeting in
Atlanta were Esther
and all experts in the field will be present (among them David Saltman
and Jean-Louis Colliot-Thelene) so we will know one way or the other
pretty soon. I sincerely hope Esther’s proof will stand the test as she
was the only one courageous enough to devote herself entirely to the
problem, regardless of slow progress.

devilish symmetries

Published April 2, 2007 by lieven

In another post we introduced
Minkowski’s question-mark function, aka the devil’s straircase
and related it to
Conways game of _contorted fractions_. Side remark : over at Good Math, Bad Math Mark Chu-Carroll is running
a mini-series on numbers&games, so far there is a post on surreal numbers,
surreal arithmetic and the connection with
games but
probably this series will go on for some time.

About a year ago I had
an email-exchange with Linas Vepstas because I was
intrigued by one of his online publications linking the fractal
symmetries of the devil’s staircase to the modular group. Unfortunately,
his paper contained some inaccuracies and I’m happy some of my comments
made it into his rewrite The Minkowski question mark, GL(2,Z) and the
modular group. Still, several
mistakes remain so read this paper only modulo his own caveat

XXXX This paper is unfinished. Although this version
corrects a number of serious errors in the previous drafts, it is still
misleading and confusing in many ways. The second half, in particular
must surely contain errors and mis-statements! Caveat emptor! XXXX

For example, on page 15 of the march 24-version he claims
that the third braid group $B_3 \simeq SL_2(\mathbb{Z}) $ which
would make life, mathematics and even physics a lot easier, but
unfortunately is not true. Recall that Artin’s defining relation for the
3-string braid group is $\sigma_1 \sigma_2 \sigma_1 = \sigma_2
\sigma_1 \sigma_2 $ as can be seen because the 3-strings below can
be transformed into each other
But from this
relation it follows that $c=(\sigma_1 \sigma_2 \sigma_1)^2 $ is
a central element in $B_3 $ and it is not difficult to verify
that indeed $B_3/ \langle c \rangle \simeq PSL_2(\mathbb{Z}) $
and $B_3/ \langle c^2 \rangle \simeq SL_2(\mathbb{Z}) $ An easy
way to see that the third braid group and the modular group are quite
different is to look at their one-dimensional representations. Any
group-map $B_3 \rightarrow \mathbb{C}^_ $ is determined by
non-zero complex numbers x and y satisfying $x^2y=y^2x $ so are
parametrized by the torus $\mathbb{C}^_ $ whereas there are only
6 one-dimensional representations of $PSL_2(\mathbb{Z}) = C_2 \ast
C_3 $ (and similarly, there are only 12 one-dimensional
$SL_2(\mathbb{Z}) $-representations). Btw. for those still
interested in noncommutative geometry : $(P)SL_2(\mathbb{Z}) $
are noncommutative manifolds whereas $B_3 $ is definitely
singular, if I ever get to the definitions of all of this… Still,
there is a gem contained in Linas’ paper and here’s my reading of it :
the fractal symmetries of the devil’s staircase form a generating
sub-semigroup $C_2 \ast \mathbb{N} $ of
$GL_2(\mathbb{Z}) $ . To begin, let us recall that the
question-mark function is defined in terms of continued fraction
expressions. So, what group of symmetries may be around the corner?
Well, if $a = \langle a_0;a_1,a_2,\ldots \rangle $ is the
continued fraction of a (see this
post for details) then if we
look at the n-th approximations $\frac{p_n}{q_n} $ (that is, the
rational numbers obtained after breaking off the continued fraction at
step n) it is failrly easy to show that $\begin{bmatrix} p_n &
p_{n-1} \\ q_n & q_{n-1} \end{bmatrix} \in GL_2(\mathbb{Z}) $ and
recall (again) that this group acts on
$\mathbb{P}^1_{\mathbb{C}} $ via Moebius transformations
$\begin{bmatrix} a & b \ c & d \end{bmatrix} $ via $z
\mapsto \frac{az+b}{cz+d} $ One of the symmetries is easy to spot
(reflexion along the 1/2-axis) That is, $?(x-1) = 1 – ?(x) $ Observe that the left-hand
side transformation is given by the Moebius transformation determined by
the matrix $r = \begin{bmatrix} -1 & 1 \\ 0 & 1 \end{bmatrix} \in
GL_2(\mathbb{Z}) $ Other symmetries are harder to see as they are
_fractal symmetries_, that is they are self-symmetries but at different
scales. For example, let us blow-up the ?-function at the interval
[1/3,1/2] and compare it with the function at the interval [1/2,1]
which has the same graph, while halving the function value. More
generally, substituting the ?-function definition using continued
fraction expressions one verifies that $?(\frac{x}{x+1}) =
\frac{1}{2} ?(x) $ and this time the left-hand transformation is
determined by the matrix $g = \begin{bmatrix} 1 & 0 \\ 1 & 1
\end{bmatrix} \in GL_2(\mathbb{Z}) $ We obtain a semi-group $S
= \langle r,g \rangle $ of fractal symmetries which are induced (the
right hand sides of the above expressions) via a 2-dimensional
representation of S $S \rightarrow GL_2(\mathbb{C})~\qquad r
\mapsto \begin{bmatrix} 1 & 0 \\ 1 & -1 \end{bmatrix}~\qquad g \mapsto
\begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{2} \end{bmatrix} $ acting
via left-multiplication on the two-dimensional vectorspace
$\mathbb{C}1+\mathbb{C}x $. We claim that S is the free
semi-group $C_2 \ast \mathbb{N} $. Clearly, $r^2=1 $ and
g is of infinite order, but we have to show that no expression of the
form $rg^{i_1}rg^{i_2}r \ldots rg^{i_l}r $ can be the identity
in S. We will prove this by computing its action on the continued
fraction expression of $a = \langle 0;a_0,a_1,\ldots \rangle $.
It is a pleasant exercise to show that $g. \langle 0;a_1,a_2,\ldots
\rangle = \langle 0;a_1+1,a_2,\ldots \rangle $ whence by induction
$g^n. \langle 0;a_1,a_2,\ldots \rangle = \langle 0;a_1+n,a_2,\ldots
\rangle $ Moreover, the action on r is given by $r. \langle
0;a_1,a_2,\ldots \rangle = \langle 0;1,a_1-1,a_2,\ldots \rangle $ if
$a_1 \not= 1 $ whereas $r. \langle 0;1,a_2,a_3,\ldots
\rangle = \langle 0;a_2+1,a_3,\ldots \rangle $ But then, as a
consequence we have that $g^{n-1}rg . \langle 0;a_1,a_2,\ldots
\rangle = \langle 0;n,a_1,a_2,\ldots \rangle $ and iterating this
procedure gives us finally that an expression $g^{j-1} r g^k r g^l
r \ldots g^z r g = (g^{j-1} r g)(g^{k-1} r g)(g^{l-1} r g) \ldots
(g^{z-1} r g) $ acts on $a = \langle 0;a_1,a_2,\ldots
\rangle $ by sending it to $\langle
0;j,k,l,\ldots,z,a_1,a_2,\ldots \rangle $ whence such an expression
can never act as the identity element, proving that indeed $S \simeq
C_2 \ast \mathbb{N} $. As for the second claim, recall from this
post that
$GL_2(\mathbb{Z}) $ is generated by the matrices $U =
\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}~\quad V = \begin{bmatrix}
0 & 1 \ -1 & 1 \end{bmatrix}~\quad R = \begin{bmatrix} 0 & 1 \ 1 & 0
\end{bmatrix} $ and a straightforward verification shows that
$r = RV,~\quad g = VU $ and $R = g^{-1}rg,~\quad
V=g^{-1}rgr,\quad U=rg^{-1}rg^2 $ whence, indeed, the semi-group S
generates the whole of $GL_2(\mathbb{Z}) $!

anabelian geometry

Published March 22, 2007 by lieven

Last time we saw
that a curve defined over $\overline{\mathbb{Q}} $ gives rise
to a permutation representation of $PSL_2(\mathbb{Z}) $ or one
of its subgroups $\Gamma_0(2) $ (of index 2) or
$\Gamma(2) $ (of index 6). As the corresponding
monodromy group is finite, this representation factors through a normal
subgroup of finite index, so it makes sense to look at the profinite
completion of $SL_2(\mathbb{Z}) $, which is the inverse limit
of finite
groups $\underset{\leftarrow}{lim}~SL_2(\mathbb{Z})/N $
where N ranges over all normalsubgroups of finite index. These
profinte completions are horrible beasts even for easy groups such as
$\mathbb{Z} $. Its profinite completion
is

$\underset{\leftarrow}{lim}~\mathbb{Z}/n\mathbb{Z} =
\prod_p \hat{\mathbb{Z}}_p $

where the right hand side
product of p-adic integers ranges over all prime numbers! The
_absolute Galois group_
$G=Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ acts on all curves
defined over $\overline{\mathbb{Q}} $ and hence (via the Belyi
maps ans the corresponding monodromy permutation representation) there
is an action of $G $ on the profinite completions of the
carthographic groups.

This is what Grothendieck calls anabelian
algebraic geometry

Returning to the general
case, since finite maps can be interpreted as coverings over
$\overline{\mathbb{Q}} $ of an algebraic curve defined over
the prime field $~\mathbb{Q} $ itself, it follows that the
Galois group $G $ of $\overline{\mathbb{Q}} $ over
$~\mathbb{Q} $ acts on the category of these maps in a
natural way.
For instance, the operation of an automorphism
$~\gamma \in G $ on a spherical map given by the rational
function above is obtained by applying $~\gamma $ to the
coefficients of the polynomials P , Q. Here, then, is that
mysterious group $G $ intervening as a transforming agent on
topologico- combinatorial forms of the most elementary possible
nature, leading us to ask questions like: are such and such oriented
maps ‚conjugate or: exactly which are the conjugates of a given
oriented map? (Visibly, there is only a finite number of these).
I considered some concrete cases (for coverings of low degree) by
various methods, J. Malgoire considered some others ‚ I doubt that
there is a uniform method for solving the problem by computer. My
reflection quickly took a more conceptual path, attempting to
apprehend the nature of this action of G.
One sees immediately
that roughly speaking, this action is expressed by a certain
outer action of G on the profinite com- pactification of the
oriented cartographic group $C_+^2 = \Gamma_0(2) $ , and this
action in its turn is deduced by passage to the quotient of the
canonical outer action of G on the profinite fundamental group
$\hat{\pi}_{0,3} $ of
$(U_{0,3})_{\overline{\mathbb{Q}}} $ where
$U_{0,3} $ denotes the typical curve of genus 0 over the
prime field Q, with three points re- moved.
This is how my
attention was drawn to what I have since termed anabelian
algebraic geometry, whose starting point was exactly a study
(limited for the moment to characteristic zero) of the action of
absolute Galois groups (particularly the groups Gal(K/K),
where K is an extension of finite type of the prime field) on
(profinite) geometric fundamental groups of algebraic varieties
(defined over K), and more particularly (break- ing with a
well-established tradition) fundamental groups which are very far
from abelian groups (and which for this reason I call
anabelian).
Among these groups, and very close to
the group $\hat{\pi}_{0,3} $ , there is the profinite
compactification of the modular group $Sl_2(\mathbb{Z}) $,
whose quotient by its centre ±1 contains the former as congruence
subgroup mod 2, and can also be interpreted as an oriented
cartographic group, namely the one classifying triangulated
oriented maps (i.e. those whose faces are all triangles or
monogons).

and a bit further, on page
250

I would like to conclude this rapid outline
with a few words of commentary on the truly unimaginable richness
of a typical anabelian group such as $SL_2(\mathbb{Z}) $
doubtless the most remarkable discrete infinite group ever
encountered, which appears in a multiplicity of avatars (of which
certain have been briefly touched on in the present report), and which
from the point of view of Galois-Teichmuller theory can be
considered as the fundamental ‚building block‚ of the
Teichmuller tower
The element of the structure of
$Sl_2(\mathbb{Z}) $ which fascinates me above all is of course
the outer action of G on its profinite compactification. By
Bielyi’s theorem, taking the profinite compactifications of subgroups
of finite index of $Sl_2(\mathbb{Z}) $, and the induced
outer action (up to also passing to an open subgroup of G), we
essentially find the fundamental groups of all algebraic curves (not
necessarily compact) defined over number fields K, and the outer
action of $Gal(\overline{K}/K) $ on them at least it is
true that every such fundamental group appears as a quotient of one
of the first groups.
Taking the anabelian yoga
(which remains conjectural) into account, which says that an anabelian
algebraic curve over a number field K (finite extension of Q) is
known up to isomorphism when we know its mixed fundamental group (or
what comes to the same thing, the outer action of
$Gal(\overline{K}/K) $ on its profinite geometric
fundamental group), we can thus say that
all algebraic
curves defined over number fields are contained in the profinite
compactification $\widehat{SL_2(\mathbb{Z})} $ and in the
knowledge of a certain subgroup G of its group of outer
automorphisms!

To study the absolute
Galois group $Gal(\overline{\mathbb{\mathbb{Q}}}/\mathbb{Q}) $ one
investigates its action on dessins denfants. Each dessin will be part of
a finite family of dessins which form one orbit under the Galois action
and one needs to find invarians to see whether two dessins might belong
to the same orbit. Such invariants are called _Galois invariants_ and
quite a few of them are known.

Among these the easiest to compute
are

the valency list of a dessin : that is the valencies of all
vertices of the same type in a dessin
the monodromy group of a dessin : the subgroup of the symmetric group $S_d $ where d is
the number of edges in the dessin generated by the partitions $\tau_0 $
and $\tau_1 $ For example, we have seen
before that the two
Mathieu-dessins

form a Galois orbit. As graphs (remeber we have to devide each
of the edges into two and the midpoints of these halfedges form one type
of vertex, the other type are the black vertices in the graphs) these
are isomorphic, but NOT as dessins as we have to take the embedding of
them on the curve into account. However, for both dessins the valency
lists are (white) : (2,2,2,2,2,2) and (black) :
(3,3,3,1,1,1) and one verifies that both monodromy groups are
isomorphic to the Mathieu simple group $M_{12} $ though they are
not conjugated as subgroups of $S_{12} $.

Recently, new
Galois invariants were obtained from physics. In Children’s drawings
from Seiberg-Witten curves
the authors argue that there is a close connection between Grothendiecks
programme of classifying dessins into Galois orbits and the physics
problem of classifying phases of N=1 gauge theories…

Apart
from curves defined over $\overline{\mathbb{Q}} $ there are
other sources of semi-simple $SL_2(\mathbb{Z}) $
representations. We will just mention two of them and may return to them
in more detail later in the course.

Sporadic simple groups and
their representations There are 26 exceptional finite simple groups
and as all of them are generated by two elements, there are epimorphisms
$\Gamma(2) \rightarrow S $ and hence all their representations
are also semi-simple $\Gamma(2) $-representations. In fact,
looking at the list of ‘standard generators’ of the sporadic
simples

(here the conjugacy classes of the generators follow the
notation of the Atlas project) we see that all but
possibly one are epimorphic images of $\Gamma_0(2) = C_2 \ast
C_{\infty} $ and that at least 12 of then are epimorphic images
of $PSL_2(\mathbb{Z}) = C_2 \ast
C_3 $.

Rational conformal field theories Another
source of $SL_2(\mathbb{Z}) $ representations is given by the
modular data associated to rational conformal field theories.

These
representations also factor through a quotient by a finite index normal
subgroup and are therefore again semi-simple
$SL_2(\mathbb{Z}) $-representations. For a readable
introduction to all of this see chapter 6 \”Modular group
representations throughout the realm\” of the
book Moonshine beyond the monster the bridge connecting algebra, modular forms and physics by Terry
Gannon. In fact, the whole book
is a good read. It introduces a completely new type of scientific text,
that of a neverending survey paper…