Tag: M-geometry

Delving into finite dimensional representations of the modular group $\mathrm{\Gamma }=PS{L}_2(\mathbb{Z})$ 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 $\mathrm{\Gamma }=PS{L}_2(\mathbb{Z})$ is the free group product $C_2\ast C_3$ it is fairly easy to see that the variety of all n-dimensional representations ${\mathbf{rep}}_n\mathrm{\Gamma }$ 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

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

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 ${\mathrm{\Gamma }}_0(6)$…

The optimistic “moonshine”-interpretation of this might be that these newforms can be viewed somehow as functions on the varieties of finite dimensional $\mathrm{\Gamma }$-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 ${\mathbf{rep}}_n\mathrm{\Gamma }$ 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 $a_i\le a_{i-1}+a_{i+1}$. 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.

Leave a Comment