Chromotopy

February 12, 2010, by Brendan Pawlowski

Here’s a neat idea I came across in a talk recently. First, a bit of background. The representation theory of a finite group over $\mathbb{C}$ , or any algebraically closed field $k$ of characteristic 0, is pretty nice. Maschke’s theorem is what makes “nice” go with “characteristic 0”: if $G$ is a finite group, $V$ is a finite-dimensional representation of $G$ over a field $k$ , and the characteristic of $k$ does not divide the order of $G$ , then $V$ is semisimple: it decomposes (uniquely!) as a direct sum $V = \bigoplus_i U_i$ , where each $U_i$ is invariant under the action of $G$ , and is irreducible, i.e. contains no proper non-trivial $G$ -invariant subspaces. So to understand the representation theory of $G$ over $k$ , it’s enough to understand the irreducible representations (and there are only finitely many of those).

On the other hand, when $k$ has characteristic $p$ dividing the order of $G$ (assume this from now on), things quickly get bad. Indeed, if $G$ is any non-cyclic $p$ -group with $p >2$ , the category of representations of $G$ over $k$ is wild (recall this is isomorphic to the category of $kG$ -modules, where $kG$ is the group algebra).

If nothing else, we can easily understand the representations of $G = \mathbb{Z}/p$ . These are modules over $k[x]/(x^p - 1) = k[x]/(x-1)^p \simeq k[y]/y^p$ ; such a thing is nothing more than a $k$ -vector space $V$ with a choice of linear map $A : V \to V$ such that $A^p = 0$ , where we only care about $A$ up to similarity, since this is what isomorphism of modules translates to where the last of these is $p \times p$ . The Jordan form of $A$ is a direct sum of some collection of the Jordan blocks $\left(\begin{array}{cc} 0 & 1\\ 0 & 0 \end{array} \right), \left(\begin{array}{ccc} 0 & 1 & 0\\ 0 & 0& 1 \\ 0 & 0 & 0 \end{array} \right), \cdots, \left(\begin{array}{cccc} 0 & 1 & \cdots & 0\\ 0 &0 &\ddots & \vdots \\ \vdots & \vdots & \ddots & 1\\ 0 & 0 & \cdots & 0\end{array} \right),$

So, representations $\mathbb{Z}/p$ are classified by integer partitions into pieces of size at most $p$ (their Jordan type), which we can write $n_1[1] + n_2[2] + \cdots + n_p[p]$ , meaning there are $n_i$ Jordan blocks of size $i$ .

Now we can get to the neat idea. Take $G$ to be an elementary $p$ -group $(\mathbb{Z}/p)^n$ , so that representations of $G$ are modules over

$kG \simeq k[x_1, \ldots, x_n]/(x_1^p - 1, \ldots, x_n^p - 1) \simeq k[y_1, \ldots, y_n]/(y_1^p, \ldots, y_n^p).$

If $P = (c_1, \ldots, c_n) \in k^n$ , notice that $\alpha_P = c_1 y_1 + \cdots + c_n y_n + 1$ has $\alpha_P^p = 1$ . The group $\langle \alpha_P \rangle$ acting on a $kG$ -module $M$ therefore makes $M$ into a representation $M_P$ of $\mathbb{Z}/p$ , and we can ask what its Jordan type is. (This can be done for more general groups $G$ , but I don’t know how).

Partially order the partitions of a fixed integer lexicographically, as usual. Then the set of points $P$ in $k^n$ such that $M_P$ does not have maximal Jordan type turns out to form a closed subvariety of $k^n$ . Unfortunately I don’t have anything exciting to say about this variety: most of the work done so far seems to be on the case when it’s empty, i.e. when $M$ has constant Jordan type, and this case is complicated enough already. It’s not even clear what partitions can arise as the (constant) Jordan type of a module; for exle, no representation of $\mathbb{Z}/p \times \mathbb{Z}/p$ can have constant Jordan type [2] + [p].

For details, see this paper of Carlson, Friedlander, and Pevtsova.