Is there a natural restatement of what a character (in the rep theoretic sense) in terms of modules? In representation theory (of finite groups over $\mathbb{C}$), most of the basics can be restated in an easier and more intuitive way using (left) $\mathbb{C}[G]$-modules. However, I don't see a natural way to define a character of a rep in terms of $\mathbb{C}[G]$-modules?
If $M$ is a (left) $\mathbb{C}[G]$-module, left multiplication by $g\in G$ defines a $\mathbb{C}[G]$-linear endomorphism (really an iso), and then we can take the trace of the corresponding matrix. Is there a more natural way to define trace in terms of $\mathbb{C}[G]$-modules than this? Should we allow this definition to extend to the trace of any $G\ni x = a_1g_1 + \cdots +  a_ng_n$, since left-multiplication by $x$ may not be an isomorphism?
 A: This is maybe a partial answer to your questions:
Let $$ be a field and let $A$ be a $$-algebra.
In the following, all occurring modules and representations are required to be finite-dimensional over $$.
Every $A$-module $M$ comes with a homomorphism of $$-algebras $ρ_M \colon A \to \mathrm{End}_(M)$, given by $ρ_M(a)(m) = am$ for all $a ∈ A$, $m ∈ M$.
This allows us to define the character of $M$ as the map
$$
  χ_M
  \colon
  A \longrightarrow  \,,
  \quad
  a \longmapsto \mathrm{tr}(ρ_M(a)) \,.
$$
This generalizes the notion of a character of a group representation:

*

*Let $G$ be a group and let $V$ be a representation of $G$ over $$. Let $M$ be the corresponding $[G]$-module.
For the resulting characters
$$
  χ_V \colon G \longrightarrow  \,,
  \quad
  χ_M \colon [G] \longrightarrow  \,,
$$
the character $χ_V$ is the restriction of $χ_M$ to $G$.

This notion of a character of an $A$-modules satisfies many properties that we are used to from characters of group representations:

*

*Isomorphic modules have the same character.

*For every short exact sequence of $A$-modules $0 \to N \to M \to P \to 0$ we have $χ_M = χ_N + χ_P$.

*This entails that $χ_{M ⊕ N} = χ_M + χ_N$ for every two $A$-modules $M$ and $N$.

*We have $χ_M(1) = \dim_(M) ⋅ 1_$ for every $A$-module $M$.

Characters of representations of groups are known to be class functions.
This can be generalized to characters of $A$-modules:

*

*Given a group $G$, every map $f'$ from $G$ to $$ extends uniquely to a linear map $f$ from $[G]$ to $G$.
The following conditions on $f$ and $F$ are equivalent:

*

*$f'$ is a class function, i.e., $f'$ is constant on conjugacy classes.

*$f'(g h g^{-1}) = f'(h)$ for all $g, h ∈ G$.

*$f'(gh) = f'(hg)$ for all $g, h ∈ G$.

*$f(ab) = f(ba)$ for all $a, b ∈ [G]$.



We therefore define a class function on $A$ to be a linear map $f$ from $A$ to $$ such that $f(ab) = f(ba)$ for all $a, b ∈ A$.
We then have the following result:

*

*The character $χ_M$ is a class function for every $A$-module $M$.

If $M$ is a simple $A$-module, then we say that the resulting character $χ_M$ is irreducible.
We then have the following results:

*

*If $\operatorname{char}() = 0$ or if $$ is algebraically closed, then the irreducible characters are linearly independent.

*If $\operatorname{char}() = 0$ and $M$ and $N$ are semisimple $A$-modules with $χ_M = χ_N$, then $M$ and $N$ are isomorphic.
(Therefore, two semisimple $A$-modules are isomorphic if and only if they have the same character.)

*If $$ is algebraically closed and $A$ is finite-dimensional and semisimple, then the irreducible characters form a basis for the space of class functions on $A$.

