As the new semester dawns, so begins my final/masters year. This includes a year-long project, for me on Grassmannians, though I’m still not sure on the details. I’ll be sharing some of the things I learn on this site, beginning with some elementary representation theory of finite groups. I didn’t study the representation theory module last year, so most of this is new to me. This article I wrote to test my memory on the things I learned from Dummit and Foote’s introduction to the theory, and therefore closely follows that exposition.
The category of -representations of a group
Working over , a representation of a finite group is a homomorphism , where is a -linear space. Hence acts on via its image in the general linear group. I’ll often write or for this action, rather than . Nonetheless it can be instructive to write out some of these ideas and arguments with the full notation.
From the perspective of category theory, a group can be described as a category with one object, and such that every arrow is an isomorphism, the arrows thus representing the elements of . Thus, we can also say that a representation is a functor , and where is the image of ‘s single object and the arrows in map to automorphisms of . With this description, it’s clear that a mapping of representations should be a natural transformation of such functors. Hence, given two representations and , a linear mapping is a morphism of representations if the following diagram commutes for all :
Surpressing the symbols and , we can write , suggesting the name “-linear” for a mapping of this sort.
Now we’ll develop these notions from a different angle, which can lead to many elegant arguments and definitions. Let be a finite group, and consider the free -vector space on the elements of . Such a space also parametrizes the set of (set-theoretic) functions : simply take any element and define for any
We can give this the structure of a (non-commutative) -algebra by identifying each element with , where , and defining a ring multiplication by setting . Multiplication for other elements is defined by forcing the distributive law. If is abelian, then this is a commutative -algebra. This ring is called .
Note that nearly always contains zerodivisors. Consider the cyclic group of order with generator . Then it is easy to see using the definitions above that is isomorphic to , where is the polynomial ring. Since is not a prime ideal, contains zerodivisors. Since any non-trivial finite group contains cyclic subgroups, the associated ring contains as a subring for some , and thus contains zerodivisors.
The category of -modules
Let be a -linear space, and let be a representation. Then can act on from the left via the action
so that this is a well-defined action. Hence induces a -module structure on .
On the other hand, let be a -module. Clearly is a -vector space, because acts via . For each , define a mapping by for all . Because has a module action on , and lies in the centre of , we have
so is linear and is clearly a homomorphism, that is, a representation of . These two preceding correspondences are inverse (up to isomorphism).
Let and be two -modules. Then a -linear map is, in particular, -linear and -linear. Thus a morphism of -modules induces a map of representations, and a mapping of representations extends via linearity to a -linear map. This establishes an equivalence between the category of -modules and that of -representations of .
This equivalence gives us practically two ways to define anything. For example, a subrepresentation can be defined as subspace such that is stable under the action of . Equivalently, a subrepresentation is a -submodule of the associated -module, since a subrepresentation must be stable under the action of both and . In the next post I’ll describe some basic results on subrepresentations.