[Verse 1]
When groups reveal their hidden face
Through matrices that dance in space
The character steps forth to see
What trace each element will be
Chi rho of g equals the trace
Of rho of g in every case
From group to complex numbers bright
This function holds the inner sight
[Chorus]
Characters are class functions, constant on each conjugacy
Orthogonal relations tell us they're perpendicular, you see
One over group order times the sum of chi i chi j star
Equals delta i j, that's the law
[Verse 2]
The character table stands complete
Square matrix where the patterns meet
Rows carry irreducible souls
Columns hold conjugacy roles
This fingerprint of representation
Captures all the information
No two groups can share this chart
It sets each structure apart
[Chorus]
Characters are class functions, constant on each conjugacy
Orthogonal relations tell us they're perpendicular, you see
One over group order times the sum of chi i chi j star
Equals delta i j, that's the law
[Bridge]
Sum dimension squared of V sub i
Must equal group order, here's why
The irreducible pieces fit
Like puzzle parts that won't admit
Any gaps or overlapping space
Each character finds its place
In orthonormal basis clean
For class functions' vector scene
[Verse 3]
From representation's complex dance
Character extracts essence at a glance
Compressed but never losing truth
This function is the living proof
That traces tell the deepest tale
When matrices might seem to fail
The character will always show
What we really need to know
[Chorus]
Characters are class functions, constant on each conjugacy
Orthogonal relations tell us they're perpendicular, you see
One over group order times the sum of chi i chi j star
Equals delta i j, that's the law
[Outro]
Chi rho maps from G to C
Trace of rho for all to see
Character table holds the key
To representation's mystery