Characters of a Group and What They Remember
In the last weeks of my Abstract Algebra 2 course, we moved from modules and group algebras to representations and finally to characters.
The setup looks like this: we start with a finite group
The character of this representation is the function
the trace of the matrix representing
One of the beautiful facts I learned from Dr. Harmon’s notes is that:
, and is constant on conjugacy classes, so it really lives on the set of conjugacy classes of .
Character tables in action
Instead of writing a static table in LaTeX, let’s play with a few small groups and their irreducible characters. Each row is an irreducible character, and each column is a conjugacy class.
Character Table Playground
Pick a finite group and explore its irreducible characters. The rows are characters, the columns are conjugacy classes.
ℤ₄ (cyclic group of order 4) • |G| = 4
Abelian group with generator a and a⁴ = e. All irreducible characters are 1-dimensional.
| χ / class | e size 1 | a size 1 | a² size 1 | a³ size 1 |
|---|---|---|---|---|
χ₀ (trivial) degree 1 | 1 | 1 | 1 | 1 |
χ₁ degree 1 | 1 | i | -1 | -i |
χ₂ degree 1 | 1 | -1 | 1 | -1 |
χ₃ degree 1 | 1 | -i | -1 | i |
Quick check: the sum of squares of the degrees equals |G|, as guaranteed by representation theory.
You can see a few important patterns immediately:
- For each group, the sum of the squares of the degrees of the irreducible characters
equals
. - For abelian groups like
, all irreducible characters are 1-dimensional. - For non-abelian examples like
and , you start seeing higher-dimensional representations and more interesting character values.
How this connects back to modules
Earlier in the course, we built the theory of
- Complex representations of
are the same as -modules. - Simple
-modules correspond to the rows in the character table. - Maschke’s Theorem (under
) tells us that every -module decomposes as a direct sum of simple modules, which shows up as decomposing characters into irreducibles.
In future posts, I’ll go back to Dr. Harmon’s “Investigations” and unpack how modules, group algebras, and tensor products build up the language needed to even state these theorems.