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 , a complex vector space , and a homomorphism

The character of this representation is the function

the trace of the matrix representing with respect to some basis of .

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 .

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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.

Choose a groupℤ₄S₃D₄

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.

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How this connects back to modules

Earlier in the course, we built the theory of -modules and tensor products. In that language:

• 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.