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Vacation reading

Published July 26, 2007 by lievenlb

Im in the process of writing/revising/extending the course notes for next year and will therefore pack more math-books than normal.

These are for a 3rd year Bachelor course on Algebraic Geometry and a 1st year Master course on Algebraic and Differential Geometry. The bachelor course was based this year partly on Miles Reid’s Undergraduate Algebraic Geometry and partly on David Mumford’s Red Book, but this turned out to be too heavy going. Next year I’ll be happy if they know enough on algebraic curves. The backbone of these two courses will be Fulton’s old but excellent Algebraic curves. It’s self contained (unlike Hartshorne’s book that assumes a prior course on commutative algebra), contains a lot of exercises and goes on to the Brill-Noether proof of Riemann-Roch. Still, Id like to extend it with the introductory chapter and the chapters on Riemann surfaces from Complex Algebraic Curves by Frances Kirwan, a bit on elliptic and modular functions from Elliptic curves by Henry McKean and Victor Moll and the adelic proof of Riemann-Roch and applications of it to the construction of algebraic codes from Algebraic curves over finite fields by Carlos Moreno. If time allows Id love to include also the chapter on zeta functions but I fear this will be difficult.

These are to spice up a 2nd year Bachelor course on Representations of Finite Groups with a tiny bit of Galois representations, both as motivation and to wet their appetite for elliptic curves and algebraic geometry. Ive received Fearless Symmetry by Avner Ash and Robert Gross only yesterday and find it hard to stop reading. It attempts to explain Galois representations and generalized reciprocity laws to a general audience and from what I read so far, they really do a terrific job. Another excellent elementary introduction to elliptic curves and Galois representations is in Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch. On a gossipy note, the appendix “The origin of the elliptic approach to Fermat’s last theorem” is fun reading. Finally, Ill also take Introduction to Fermat’s Last Theorem by Alf van der Poorten along simply because I love his writing style.

These are included just for fun. The Poincare Conjecture by Donal O’Shea because I know far too little about it, Letters to a Young Mathematician by Ian Stewart because I like the concept of the book and finally The sensual (quadratic) form by John Conway because I need to have at all times at least one Conway-book nearby.

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Hexagonal Moonshine (3)

Published July 13, 2007 by lievenlb

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

Hexagonal Moonshine (2)

Published July 11, 2007 by lievenlb

Delving into finite dimensional representations of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ it is perhaps not too surprising to discover numerical connections with modular functions. Here, one such strange observation and a possible explanation.

Using the _fact_ that the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is the free group product $C_2 \ast C_3 $ it is fairly easy to see that the variety of all n-dimensional representations $\mathbf{rep}_n~\Gamma $ is smooth (though it contains several connected components). Some of these components will contain simple representations, some will not. Anyway, we are not interested in all n-dimensional representations but in the isomorphism classes of such representations. The best algebraic approximation to this problem is by studying the quotient varieties

$\mathbf{iss}_n~\Gamma = \mathbf{rep}_n~\Gamma // GL(n) $

under the action of $GL(n) $ by basechange. Geometric invariant theory tells us that the points of this quotient variety correspond to isoclasses of semi-simple n-dimensional representations (whence the notation $\mathbf{iss}_n $). Again, these quotient varieties split into several connected components, some of which will have an open subset of points corresponding to simple representations.

It is a natural idea to compute the codimension of subvariety of proper semi-simples in the component of maximal dimension containing simple representations. _M-geometry_ allows you to reduce such calculation to quiver-problems. Anyway, if one does this for small values of n one obtains the following sequence of codimension-numbers (starting with $n=1 $

0,1,1,1,1,3,1,3,3,3,3,5,3,5,5,5,5,7,5,7,…

which doesnt seem too exciting before you feed it to Sloan’s integer sequences encyclopedia when one discovers that it is precisely sequence A063195 which gives the dimensions of weight 2n _cuspidal newforms_
for $\Gamma_0(6) $…

The optimistic “moonshine”-interpretation of this might be that these newforms can be viewed somehow as functions on the varieties of finite dimensional $\Gamma $-representations having the property that they pick out generic simple representations as their non-zeroes.

Be that as it may (one never knows in these matters), here a more down-to-earth explanation. The sequence A063195 obviously has a 6-periodicity behaviour so it suffices to understand why the codimension-sequence should have a similar feature (modulo computing the first few terms of it and observing the coincidence with the first few terms of A063195).

The modular group has exactly 6 one-dimensional representations and if one calculates their clan as in hexagonal moonshine (1) one obtains the hexagonal quiver

[tex]\xymatrix{& \vtx{S_1} \ar@/^/[dl] \ar@/^/[dr] & \\ \vtx{S_6} \ar@/^/[ur] \ar@/^/[d] & & \vtx{S_2} \ar@/^/[ul] \ar@/^/[d] \\ \vtx{S_5} \ar@/^/[u] \ar@/^/[dr] & & \vtx{S_3} \ar@/^/[u] \ar@/^/[dl] \\ & \vtx{S_4} \ar@/^/[ur] \ar@/^/[ul] & }[/tex]

M-geometry tells us that this clan contains enough information to determine the components of $\mathbf{rep}_n~\Gamma $ that contain simple representations. They correspond to dimension-vectors of this hexagonal quiver, say

$~(a_1,a_2,a_3,a_4,a_5,a_6) $

such that $a_i \leq a_{i-1}+a_{i+1} $. Moreover, the component is of maximal dimension if the components $a_i $ are evenly spread over the six vertices.
This then explains that the codimension sequence we are interested in must satisfy 6-periodicity.

Reference

This post corrects the erroneous statement made in math.AG/0610587 that the codimension sequence are the dimensions of weight 2n modular forms. The day the paper hit the arXiv I informed the author of the mistakes he made and told him how they could be corrected. Having waited 9 months I’ve given up hope that a revision/correction is imminent.

Tag: representations

Vacation reading

Hexagonal Moonshine (3)

Hexagonal Moonshine (2)