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, $G = PSL_2(\mathbb{F}_7)$ also known as

Klein\’s group. John Baez has written a [beautiful page](http://math.ucr.edu/home/baez/klein.html) on the Klein quartic and

its symmetries. Another useful source of information is a paper by Noam

Elkies [The Klein quartic in number theory](www.msri.org/publications/books/Book35/files/elkies.pd).

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](www.matrix.ua.ac.be/master/coursenotes/onequiver.pdf) of this

qurve has the following shape $\xymatrix{\vtx{} \ar@/^/[dd] \\

\\ \vtx{} \ar@/^/[uu]} $ $\xymatrix{& \vtx{} \ar[ddl] & \\

& & \\ \vtx{} \ar[rr] & & \vtx{} \ar[uul]} $ $\xymatrix{& &

\vtx{} \ar[dll] & & \\ \vtx{} \ar[d] & & & & \vtx{} \ar[ull] \\ \vtx{}

\ar[dr] & & & & \vtx{} \ar[u] \\ & \vtx{} \ar[rr] & & \vtx{} \ar[ur]

&} $ 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).

# the Klein stack

Published in featured

## Comments