Chromotopy

Watercooler research

Jordan type of a modular representation

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 , or any algebraically closed field of characteristic 0, is pretty nice. Maschke’s theorem is what makes “nice” go with “characteristic 0”: if is a finite group, is a finite-dimensional representation of over a field , and the characteristic of does not divide the order of , then is *semisimple*: it decomposes (uniquely!) as a direct sum , where each is invariant under the action of , and is irreducible, i.e. contains no proper non-trivial -invariant subspaces. So to understand the representation theory of over , it’s enough to understand the irreducible representations (and there are only finitely many of those).

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

If nothing else, we can easily understand the representations of . These are modules over ; such a thing is nothing more than a -vector space with a choice of linear map such that , where we only care about up to similarity, since this is what isomorphism of modules translates to where the last of these is . The Jordan form of is a direct sum of some collection of the Jordan blocks

So, representations are classified by integer partitions into pieces of size at most (their *Jordan type*), which we can write , meaning there are Jordan blocks of size .

Now we can get to the neat idea. Take to be an elementary -group , so that representations of are modules over

If , notice that has . The group acting on a -module therefore makes into a representation of, and we can ask what its Jordan type is. (This can be done for more general groups , but I don’t know how).

Partially order the partitions of a fixed integer lexicographically, as usual. Then the set of points in such that does not have maximal Jordan type turns out to form a closed subvariety of . 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 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 can have constant Jordan type [2] + [p].

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