# Thread: Modules over group algebra

1. ## Modules over group algebra

I'm having a little trouble understanding (intuitively or otherwise) what is meant by a module over a group algebra FG (F a field, G a finite group) and how it relates to transformations.

Identify $\displaystyle S_3$ with the subgroup of $\displaystyle S_4$ which fixes 4 ($\displaystyle S_n$ is of course the symmetric group on $\displaystyle n$ letters). Then $\displaystyle S_3$ acts on $\displaystyle S_4$ by conjugation (???). Find the character of the linearization $\displaystyle \mathbb{C} S_4$ as a $\displaystyle \mathbb{C} S_3$-module, and determine the multiplicities of each simple $\displaystyle \mathbb{C} S_3$-module in $\displaystyle \mathbb{C} S_4$.

I would be very very grateful for any light anyone could shed on this question; especially any general comments about FG-modules and representations.

2. Originally Posted by AtticusRyan
I'm having a little trouble understanding (intuitively or otherwise) what is meant by a module over a group algebra FG (F a field, G a finite group) and how it relates to transformations.

Identify $\displaystyle S_3$ with the subgroup of $\displaystyle S_4$ which fixes 4 ($\displaystyle S_n$ is of course the symmetric group on $\displaystyle n$ letters). Then $\displaystyle S_3$ acts on $\displaystyle S_4$ by conjugation (???). Find the character of the linearization $\displaystyle \mathbb{C} S_4$ as a $\displaystyle \mathbb{C} S_3$-module, and determine the multiplicities of each simple $\displaystyle \mathbb{C} S_3$-module in $\displaystyle \mathbb{C} S_4$.

I would be very very grateful for any light anyone could shed on this question; especially any general comments about FG-modules and representations.

to get you started:

well, a (finite dimensional) vector space $\displaystyle V$ over a field $\displaystyle F$ is called an $\displaystyle FG$ module ($\displaystyle G$ is a group usually finite) or a $\displaystyle G$ module if there exists a homomorphism $\displaystyle \rho: G \longrightarrow \text{GL}(V).$

in your problem: $\displaystyle F=\mathbb{C}, \ G=S_3, \ V=\mathbb{C}S_4.$ the homomorphism $\displaystyle \rho$ is defined by $\displaystyle \rho(\sigma)(\alpha)=\sigma \alpha \sigma^{-1},$ for all $\displaystyle \sigma \in S_3, \ \alpha \in \mathbb{C}S_4.$ since the value of $\displaystyle \chi,$ the character corresponding

to the representation $\displaystyle \rho,$ over a conjugacy class is constant, you only need to find $\displaystyle \chi(\sigma_j)=\text{Tr}(\rho(\sigma_j)), \ j=1,2,3,$ where $\displaystyle \sigma_1=1_{S_3}, \ \sigma_2=(1 \ 2), \ \sigma_3=(1 \ 2 \ 3).$

3. I think I understand; could you perhaps help me find the matrix $\displaystyle \rho(1_{S_3})$? In particular what is our basis of $\displaystyle V$? This is the kind of stuff I don't fully understand how to do.

Obviously $\displaystyle \chi(1_{S_3})=\dim(V)$ but what is the dimension of $\displaystyle V$ as a $\displaystyle \mathbb{C}S_3$-module?

4. Originally Posted by AtticusRyan
Identify $\displaystyle S_3$ with the subgroup of $\displaystyle S_4$ which fixes 4 ($\displaystyle S_n$ is of course the symmetric group on $\displaystyle n$ letters). Then $\displaystyle S_3$ acts on $\displaystyle S_4$ by conjugation (???). Find the character of the linearization $\displaystyle \mathbb{C} S_4$ as a $\displaystyle \mathbb{C} S_3$-module, and determine the multiplicities of each simple $\displaystyle \mathbb{C} S_3$-module in $\displaystyle \mathbb{C} S_4$.
This is my attempt. Hope it helps.

Lemma 1. Let N be a normal subgroup of G and let U be a $\displaystyle \mathbb{C}(G/N)$-module. Then U admits a canonical $\displaystyle \mathbb{C}G$-module structure, with a subspace of U being a $\displaystyle \mathbb{C}G$-submodule iff it is a $\displaystyle \mathbb{C}(G/N)$-submodule. If $\displaystyle \psi$ is the character of the $\displaystyle \mathbb{C}(G/N)$-module U, then the character of the $\displaystyle \mathbb{C}G$-module U is $\displaystyle \psi \cdot \mu$, where $\displaystyle \mu:G \rightarrow G/N$ is the natural map.

Lemma 1 says that given $\displaystyle g \in G$, $\displaystyle u \in U$ and $\displaystyle N \triangleleft G$, we can define gu = (gN)u such that the linear transformation of U induced by g under the action of G on U is the same as that induced by gN under the action of G/N on U. This gives U a $\displaystyle \mathbb{C}G$-module structure where $\displaystyle \mathbb{C}G$-submodule of U are exactly the $\displaystyle \mathbb{C}(G/N)$-submodule of U.

Let N={1, (1 2)(3 4), (13)(24), (14)(23)}. Then $\displaystyle S_4/N \cong S_3$, where $\displaystyle N \triangleleft S_4$. Now we lift the characters of $\displaystyle S_3$ to the characters of $\displaystyle S_4$.

The character table of $\displaystyle S_3 \cong S_4/N$ is

$\displaystyle \begin{matrix} & N & (12)N & (123)N\\ \phi^1 (trivial) & 1 & 1 & 1\\ \phi^2 (sign) & 1 & -1 & 1\\ \phi^3 (permutation) & 2 &0 & -1 \end{matrix}$

Since $\displaystyle S_3$ have three conjugacy classes, there are three irreducible characters in $\displaystyle S_3$. Note that characters are constant in conjugacy classes.

Now we use the lifting property and find the character table of $\displaystyle S_4$. The character table of $\displaystyle S_4$ is

$\displaystyle \begin{matrix} & 1 & (12) & (123) & (1234) & (12)(34)\\ \chi^1 (trivial) & 1 & 1 & 1 & 1 & 1\\ \chi^2 (sign) & 1 & -1 & 1 & -1 & 1\\ \chi^3 (permutation) & 2 &0 & -1 & 0 & 2\\ \cdots & \cdots& \cdots& \cdots& \cdots& \cdots\\ \end{matrix}$

The fourth and fifth column is simply acquired by using the lifting property that $\displaystyle \chi^i(1234) = \phi^i((12)N)$ and $\displaystyle \chi^i((12)(34)) = \phi^i(N)$.

While $\displaystyle S_3$ has three conjugacy classes, $\displaystyle S_4$ has five conjugacy classes. This implies that $\displaystyle S_4$ has five irreducible characters and two of them cannot be lifted from $\displaystyle S_3$. To get $\displaystyle \chi^4$, suppose $\displaystyle S_4$ act transitively on the set X ={1, 2, 3, 4}; let $\displaystyle \pi$ be the character of the $\displaystyle \mathbb{C}S_4$-module $\displaystyle \mathbb{C}X$. $\displaystyle \pi(g)$ equals the number of fixed points under the action of g on X, and we have $\displaystyle \chi^4 = \pi - \chi^1$ (I'll leave it to you to verify that $\displaystyle \chi^4$ is irreducible). Using the fact that $\displaystyle \chi^2\chi^4$ is irreducible, the last irreducible character is $\displaystyle \chi^5 = \chi^2\chi^4$.

Anyhow, I was not able to figure out the multiplicity of $\displaystyle \mathbb{C}S_3$-module in $\displaystyle \mathbb{C}S_4$.