The Group Ring and the Regular Representation

In the previous post, we saw how to decompose a given group representation into irreducibles. But we still don’t know much about the irreducible representations of a (finite) group. What do they look like? How many are there? Infinitely many?

In this post, we’ll construct the group ring of a group. Treating this as a vector space, we get the regular representation, which turns out to contain all the irreducible representations of $G$!

The group ring $FG$

Given a (finite) group $G$ and a field $F$, we can treat each element of $G$ as a basis element of a vector space over $F$. The resulting vector space generated by $g \in G$ is

\[FG := \left\{\sum_{g\in G} \alpha_g g: \alpha_g \in F \right\}.\]

Let’s do this is Sage with the group $G = D_4$ and the field $F = \mathbb{Q}$:

(The Sage cells in this post are linked, so things may not work if you don’t execute them in order.)

We can view $v \in FG$ as vector in $F^n$, where $n$ is the size of $G$ :

Here, we’re treating each $g \in G$ as a basis element of $FG$

Vectors in $FG$ are added component-wise:

\[\left(\sum_{g \in G} \alpha_g g\right) + \left(\sum_{g\in G} \beta_g g\right) = \sum_{g \in G} (\alpha_g+\beta_g) g.\]

Multiplication as a linear transformation

In fact $FG$ is also a ring (called the group ring), because we can multiply vectors using the multiplication rule of the group $G$:

\[\left(\sum_{h \in G} \alpha_h h\right) \left(\sum_{g\in G} \beta_g g\right) = \sum_{h,g \in G} (\alpha_h \beta_g) hg.\]

That wasn’t very illuminating. However, treating multiplication by $v \in FG$ as a function

\[\begin{align*} T_v: FG &\to FG \\ w &\mapsto vw, \end{align*}\]

one can check that each $T_v$ is a linear transformation! We can thus represent $T_v$ as a matrix whose columns are $T_v(g), g \in G$:

The regular representation

We’re especially interested in $T_g, g \in G$. These are invertible, with inverse $T_{g^{-1}}$, and their matrices are all permutation matrices, because multiplying by $g \in G$ simply permutes elements of $G$:

Define a function $\rho_{FG}$ which assigns to each $g\in G$ the corresponding $T_g$:

\[\begin{align*} \rho_{FG}: G &\to \mathrm{GL}(FG) \\ g &\mapsto T_g \end{align*}\]

Then $(FG,\rho_{FG})$ is the regular representation of $G$ over $F$.

The regular representation of any non-trivial group is not irreducible. In fact, it is a direct sum of all the irreducible representations of $G$! What’s more, if $(V,\rho)$ is an irreducible representation of $G$ and $\dim V = k$, then $V$ occurs $k$ times in the direct-sum decomposition of $FG$!

Let’s apply the decomposition algorithm in the previous post to $(FG,\rho_{FG})$ (this might take a while to run):

So the regular representation of $D_4$ decomposes into four (distinct) $1$-dim representations and two (isomorphic) $2$-dim ones.

Building character

We’ve spent a lot of time working directly with representations of a group. While more concrete, the actual matrix representations themselves tend to be a little clumsy, especially when the groups in question get large.

In the next few posts, I’ll switch gears to character theory, which is a simpler but more powerful way of working with group representations.

Written on February 15, 2015