# The Unapologetic Mathematician

## Modules

With the group algebra in hand, we now define a “$G$-module” to be a module for the group algebra of $G$. That is, it’s a (finite-dimensional) vector space $V$ and a bilinear map $A:\mathbb{C}[G]\times V\to V$. This map must satisfy $A(\mathbf{e},v)=v$ and $A(\mathbf{g},A(\mathbf{h},v))=A(\mathbf{gh},v)$.

This is really the same thing as a representation, since we may as well pick a basis $\{e_i\}$ for $V$ and write $V=\mathbb{C}^d$. Then for any $g\in G$ we can write

$\displaystyle A(\mathbf{g},e_i)=\sum\limits_{j=1}^dm_i^je_j$

That is, $A(\mathbf{g},\underbar{\hphantom{X}})$ is a linear map from $V$ to itself, with its matrix entries given by $m_i^j$. We define this matrix to be $\rho(g)$, which must be a representation because of the conditions on $A$ above.

Conversely, if we have a matrix representation $\rho:G\to GL_d$, we can define a module map for $\mathbb{C}^d$ as

$\displaystyle A(\mathbf{g},v)=\rho(g)v$

where we apply the matrix $\rho(g)$ to the column vector $v$. This must satisfy the above conditions, since they reflect the fact that $\rho$ is a representation.

In fact, to define $A$, all we really need to do is to define it for the basis elements $\mathbf{g}\in\mathbb{C}[G]$. Then linearity will take care of the rest of the group algebra. That is, we can just as well say that a $G$-module is a vector space $V$ and a function $A:G\times V\to V$ satisfying the following three conditions:

• $A$ is linear in $V$: $A(g,cv+dw)=cA(g,v)+dA(g,w)$.
• $A$ preserves the identity: $A(e,v)=v$.
• $A$ preserves the group operation: $A(g,A(h,v))=A(gh,v)$.

The difference between the representation viewpoint and the $G$-module viewpoint is that representations emphasize the group elements and their actions, while $G$-modules emphasize the representing space $V$. This viewpoint will be extremely helpful when we want to consider a representation as a thing in and of itself. It’s easier to do this when we think of it as a vector space equipped with the extra structure of a $G$-action.

September 15, 2010