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Hexagonal Moonshine (3)

Hexagons keep on popping up in the representation theory of the modular group and its close associates. We have seen before that singularities in 2-dimensional representation varieties of the three string braid group $B_3 $ are ‘clanned together’ in hexagons and last time Ive mentioned (in passing) that the representation theory of the modular group is controlled by the double quiver of the extended Dynkin diagram $\tilde{A_5} $, which is an hexagon…

Today we’re off to find representations of the extended modular group $\tilde{\Gamma} = PGL_2(\mathbb{Z}) $, which is obtained by adding to the modular group (see this post for a proof of generation)

$\Gamma = \langle U=\begin{bmatrix} 0 & -1 \\\ 1 & 0 \end{bmatrix},V=\begin{bmatrix} 0 & 1 \\\ -1 & 1 \end{bmatrix} \rangle $ the matrix $R=\begin{bmatrix} 0 & 1 \\\ 1 & 0 \end{bmatrix} $

In terms of generators and relations, one easily verfifies that

$\tilde{\Gamma} = \langle~U,V,R~|~U^2=R^2=V^3=(RU)^2=(RV)^2=1~\rangle $

and therefore $\tilde{\Gamma} $ is the amalgamated free product of the dihedral groups $D_2 $ and $D_3 $ over their common subgroup $C_2 = \langle~R~\rangle $, that is

$\tilde{\Gamma} = \langle U,R | U^2=R^2=(RU)^2=1 \rangle \ast_{\langle R | R^2=1 \rangle} \langle V,R | V^3=R^2=(RV)^2=1 \rangle = D_2 \ast_{C_2} D_3 $

From this description it is easy to find all n-dimensional $\tilde{\Gamma} $-representations $V $ and relate them to quiver-representations. $D_2 = C_2 \times C_2 $ and hence has 4 1-dimensonal simples $S_1,S_2,S_3,S_4 $. Restricting $V\downarrow_{D_2} $ to the subgroup $D_2 $ it decomposes as

$V \downarrow_{D_2} \simeq S_1^{\oplus a_1} \oplus S_2^{\oplus a_2} \oplus S_3^{\oplus a_3} \oplus S_4^{\oplus a_4} $ with $a_1+a_2+a_3+a_4=n $

Similarly, because $D_3=S_3 $ has two one-dimensional representations $T,S $ (the trivial and the sign representation) and one simple 2-dimensional representation $W $, restricting $V $ to this subgroup gives a decomposition

$V \downarrow_{D_3} \simeq T^{b_1} \oplus S^{\oplus b_2} \oplus W^{\oplus b_3} $, this time with $b_1+b_2+2b_3=n $

Restricting both decompositions further down to the common subgroup $C_2 $ one obtains a $C_2 $-isomorphism $V \downarrow_{D_2} \rightarrow^{\phi} V \downarrow_{D_3} $ which implies also that the above numbers must be chosen such that $a_1+a_3=b_1+b_3 $ and $a_2+a_4=b_2+b_3 $. We can summarize all this info about $V $ in a representation of the quiver

Here, the vertex spaces on the left are the iso-typical factors of $V \downarrow_{D_2} $ and those on the right those of $V \downarrow_{D_3} $ and the arrows give the block-components of the $C_2 $-isomorphism $\phi $. The nice things is that one can also reverse this process to get all $\tilde{\Gamma} $-representations from $\theta $-semistable representations of this quiver (having the additional condition that the square matrix made of the arrows is invertible) and isomorphisms of group-representation correspond to those of quiver-representations!

This proves that for all n the varieties of n-dimensional representations $\mathbf{rep}_n~\tilde{\Gamma} $ are smooth (but have several components corresponding to the different dimension vectors $~(a_1,a_2,a_3,a_4;b_1,b_2,b_3) $ such that $\sum a_i = n = b_1+b_2+2b_3 $.

The basic principle of _M-geometry_ is that a lot of the representation theory follows from the ‘clan’ (see this post) determined by the simples of smallest dimensions. In the case of the extended modular group $\tilde{\Gamma} $ it follows that there are exactly 4 one-dimensional simples and exactly 4 2-dimensional simples, corresponding to the dimension vectors

$\begin{cases} a=(0,0,0,1;0,1,0) \\\ b=(0,1,0,0;0,1,0) \\\ c=(1,0,0,0;1,0,0) \\\ d=(0,0,1,0;1,0,0) \end{cases} $ resp. $\begin{cases} e=(0,1,1,0;0,0,1) \\\ f=(1,0,0,1;0,0,1) \\\ g=(0,0,1,1;0,0,1) \\\ h=(1,1,0,0;0,0,1) \end{cases} $

If one calculates the ‘clan’ of these 8 simples one obtains the double quiver of the graph on the left. Note that a and b appear twice, so one should glue the left and right hand sides together as a Moebius-strip. That is, the clan determining the representation theory of the extended modular group is a Moebius strip made of two hexagons!

However, one should not focuss too much on the hexagons (that is, the extended Dynkin diagram $\tilde{A_5} $) here. The two ‘backbones’ (e–f and g–h) have their vertices corresponding to 2-dimensional simples whereas the topand bottom vertices correspond to one-dimensional simples. Hence, the correct way to look at this clan is as two copies of the double quiver of the extended Dynkin diagram $\tilde{D_5} $ glued over their leaf vertices to form a Moebius strip. Remark that the components of the sotropic root of $\tilde{D_5} $ give the dimensions of the corresponding $\tilde{\Gamma} $ simples.

The remarkable ubiquity of (extended) Dynkins never ceases to amaze!

One Comment

group think

The
moment I read about it, I ordered the book, but received at least three
emails from Amazon.co.uk apologizing for not being able to find me a
copy of Lee Smolin’s The trouble with physics.
A very
considerate review of the book can be found at Background Independence, Christine Dantas’
old blog. Btw. I’m happy Christine has set up a new blog called
Theorema Egregium. Here’s the section
in her review that convinced me to have a look at the book myself.

I do not wish to make public some of my old, deep own
feelings about what I think science is and how it should be conducted.
There are of course certain points that I often do make public, but
there are some others that tormented me for quite a long time now, and
are so personal and even of emotive nature that I would rather keep them
to myself. But this is the fact per se that should be mentioned here,
since this is the contribution that I feel I can give on examining his
book: I found out that he was addressing some of my personal views and
doubts, of course from his own perspective and wisdom, but it was like
talking to an old friend who followed my own career in science and
understood what troubled me most for all those years. So this book is
for you if you want to be challenged over your own vision of science and
how you fit in it.

Finally, after all these months, just
before going on vacation I discovered a copy in one of my favourite
bookshops in Antwerp and took it along. I dont know Christine’s
favourite chapters of the book but I feel somehow I’ll be not too far
off mark in believing that chapter 16 “How Do You Fight Sociology?” will
be among them. This chapter (just 27 pages) should be read and reread by
all scientists. In it, Lee Smolin describes community behaviour of
certain scientific groups (he has the stringtheory-community in mind but
I’m sure anyone will recognise some of its behavior in groups closer to
ones own research-interests. I certainly did…). Here we go (citing
from page 284)

1. _Tremendous self-confidence_ ,
leading to a sense of entitlement and of belonging to an elite of
experts.
2. _An unusually monolithic community_ , with a
strong sense of consensus, whether driven by evidence or not, and an
unusual uniformity of views on open questions. These views seem related
to the existence of a hierarchical structure in which ideas of a few
leaders dictate the viewpoint, strategy, and direction of the field,
3. In some cases, a _sense of identification with the group_ ,
akin to identification with a religious faith or political platform.
4. A strong sense of the _boundary between the group and other
experts_ .
5. A _disregard for and disinterest in_ the
ideas, opinions, and work of experts who are not part of the group, and
a preference for talking only with other members of the commnity.
6. A tendency to _interpret evidence optimistically_ , to believe
exaggerated or incorrect statements of results, and to disregard the
possibility that the theory might be wrong. This is coupled with a
tendency to _believe results are true because they are widely
believed’_ even if one has not checked (or even seen) the proof
oneself.
7. A lack of appreciation for the extent to which a
research program ought to involve risk.

Although spotting
such behaviour can be depressing and/or frustrating, Smolin’s analysis
is that such groups are doomed to perish sooner or later for it is
exactly the kind of behaviour sociologists and psychologists recognize
as groupthink, following the Yale psychologist Irving Janis, “a mode
of thinking that people engage in when they are deeply involved in a
cohesive in-group, when the members’ strivings for unanimity override
their motivation to realistically appraise alternative courses of
action”. Groupthink is responsable for failures of decision making by
groups of experts such as the “failure of NASA to prevent the Challenger
disaster, the failure of the West to anticipate the collapse of the
Soviet Union, the failure of the American automobile companies to
feresee the demand for smaller cars, and most recently – and perhaps
most calamitously – the Bush administration’s rush to war on the basis
of a false belief that Iraq had weapons of mass destruction.” (Smolin,
page 286). An aspect of these groupthinking science-groups that worries
me most of all is their making of exagerated claims to potential
applications, not supported (yet) by solid proof. Short-time effect may
be to attract more people to the subject and to keep doubting followers
on board, but in the long term (at least if the claimed results remain
out of reach) this will destroy the subject itself (and, sadly enough,
also closeby subjects making no outrageous claims…). My advice to
people making such claims is : do a Perelman! Rather than doing a
PR-job, devote yourself for as long as it takes to prove your hopes,
somewhere in splendid isolation and come back victoriously. I have a
spare set of keys if you are in search for the perfect location…

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