Klein’s quartic $X$is the smooth plane projective curve defined by $x^3y+y^3z+z^3x=0$ and is one of the most remarkable mathematical objects around. For example, it is a Hurwitz curve meaning that the finite group of symmetries (when the genus is at least two this group can have at most $84(g-1)$ elements) is as large as possible, which in the case of the quartic is $168$ and the group itself is the unique simple group of that order, $G = PSL2(\mathbb{F}7)$ also known as Klein\’s group. John Baez has written a beautiful page on the Klein quartic and its symmetries. Another useful source of information is a paper by Noam Elkies The Klein quartic in number theory.
The quotient map $X \rightarrow X/G \simeq \mathbb{P}^1$ has three branch points of orders $2,3,7$ in the points on $\mathbb{P}^1$ with coordinates $1728,0,\infty$. These points correspond to the three non-free $G$-orbits consisting resp. of $84,56$ and $24$ points.
Now, remove from $X$ a couple of $G$-orbits to obtain an affine open subset $Y$ such that $G$ acts on its cordinate ring $\mathbb{C}[Y]$ and form the Klein stack (or hereditary order) $\mathbb{C}[Y] \bigstar G$, the skew group algebra. In case the open subset $Y$ contains all non-free orbits, the one quiver of this qurve has the following shape Here, the three components correspond to the three non-free orbits and the vertices correspond to the isoclasses of simple $\mathbb{C}[Y] \bigstar G$ of dimension smaller than $168$. There are two such of dimension $84$, three of dimension $56$ and seven of dimension $24$ which I gave the non-imaginative names \’twins\’, \’trinity\’ and \’the dwarfs\’. As we want to spice up later this Klein stack to a larger group, we need to know the structure of these exceptional simples as $G$-representations. Surely, someone must have written a paper on the general problem of finding the $G$-structure of simples of skew-group algebras $A \bigstar G$, so if you know a reference please let me know. I used an old paper by Idun Reiten and Christine Riedtmann to do this case (which is easier as the stabilizer subgroups are cyclic and hence the induced representations of their one-dimensionals correspond to the exceptional simples).

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Posted in geometry
Written on Wed, 15 June 2005 at 12:40 pm
Tags: Elkies, groups, Klein, representations, simples
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