Getting into my project with a little representation theory

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 \C-representations of a group G

Working over \C, a representation of a finite group G is a homomorphism \rho\colon G \to \GL{(V)}, where V is a \C-linear space. Hence G acts on V via its image in the general linear group. I’ll often write gv or g\cdot v for this action, rather than \rho(g)(v). 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 G with one object, and such that every arrow is an isomorphism, the arrows thus representing the elements of G. Thus, we can also say that a representation is a functor G \to \mathsf{Vect}_\C, and where V is the image of G‘s single object and the arrows in G map to automorphisms of V. With this description, it’s clear that a mapping of representations should be a natural transformation of such functors. Hence, given two representations \rho\colon G \to \GL{(V)} and \sigma \colon G \to \GL{(W)}, a linear mapping \varphi\colon V \to W is a morphism of representations if the following diagram commutes for all g:

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Surpressing the symbols \rho and \sigma, we can write \varphi(g\cdot v) = g\cdot \varphi(v), suggesting the name “G-linear” for a mapping of this sort.

The ring \C G

Now we’ll develop these notions from a different angle, which can lead to many elegant arguments and definitions. Let G be a finite group, and consider the free \C-vector space on the elements of G. Such a space also parametrizes the set of (set-theoretic) functions G \to \C: simply take any element \sum \alpha_i g_i and define for any g_j

    \[\left(\sum \alpha_i g_i \right)(g_j) = \alpha_j.\]

We can give this the structure of a (non-commutative) \C-algebra by identifying each element \alpha \in \C with \alpha g_1, where g_1 = e_G, and defining a ring multiplication by setting (\alpha g_i)(\beta g_j) = (\alpha \beta)(g_ig_j). Multiplication for other elements is defined by forcing the distributive law. If G is abelian, then this is a commutative \C-algebra. This ring is called \C G.

Note that \C G nearly always contains zerodivisors. Consider the cyclic group C_n of order n with generator g. Then it is easy to see using the definitions above that \C C_n is isomorphic to \C[g]/(g^n - 1), where \C[g] is the polynomial ring. Since (g^n -1) is not a prime ideal, \C C_n contains zerodivisors. Since any non-trivial finite group G contains cyclic subgroups, the associated ring \C G contains \C C_n as a subring for some n, and thus contains zerodivisors.

The category of \C G-modules

Let V be a \C-linear space, and let \rho\colon G \to \GL{(V)} be a representation. Then \C G can act on V from the left via the action

    \[\left(\sum \alpha_i g_i\right)(v) = \sum \alpha( g_i \cdot v).\]

Note that

    \[(g_i g_j)\cdot (v) = \rho(g_i g_j)(v) = \rho(g_i)\circ \rho(g_j)(v) = \rho(g_i)(\rho(g_j)(v)) = g_i\cdot(g_j \cdot v),\]

so that this is a well-defined action. Hence \rho induces a \C G-module structure on V.

On the other hand, let M be a \C G-module. Clearly M is a \C-vector space, because \C acts via \C G. For each g \in G, define a mapping \rho \colon G \to \GL{(V)} by \rho(g) (m) = g\cdot m for all m \in M. Because g has a module action on M, and \C lies in the centre of \C G, we have

     \begin{align*} \rho(g)(\alpha m_1 + \beta m_2) &= g(\alpha m_1 + \beta m_2) \\ &= \alpha g  m_1 + \beta g m_2 \\ &= \alpha \rho(g)(m_1) + \beta \rho(g)(m_2), \end{align*}

so \rho(g) is linear and \rho is clearly a homomorphism, that is, a representation of G. These two preceding correspondences are inverse (up to isomorphism).

Let V and W be two \C G-modules. Then a \C G-linear map V \to W is, in particular, \C-linear and G-linear. Thus a morphism of \C G-modules induces a map of representations, and a mapping of representations extends via linearity to a \C G-linear map. This establishes an equivalence between the category of \C G-modules and that of \C-representations of G.

This equivalence gives us practically two ways to define anything. For example, a subrepresentation can be defined as subspace U \subset V such that U is stable under the action of G. Equivalently, a subrepresentation is a \C G-submodule of the associated \C G-module, since a subrepresentation must be stable under the action of both \C and G. In the next post I’ll describe some basic results on subrepresentations.

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