Decomposing a Representation from Its Character
Build the character table of a finite group and use it to read off how any representation breaks into irreducibles.
What a learner can do afterwards
- Construct the character table of a small non-abelian group
- Decompose a permutation representation by taking inner products with the irreducible characters
- Read a normal subgroup off the table from the kernel of a character
1 · Read
A character table lists every irreducible character of a group. Each row is one irreducible character and each column is one conjugacy class, so each entry is that character's value there.
To break a representation into irreducibles, take an inner product with each row. Multiply the matching entries and add, the way a dot product multiplies matching components and adds. The result is a whole number: how many copies of that irreducible sit inside.
Take the symmetries of a triangle and their shuffling of the three corners. Its character dotted with the all-ones row gives 1, so one copy of the trivial piece sits inside, and the remaining pieces are read off the same way.
To spot a normal subgroup, look at one row's kernel: the elements whose value equals the row's dimension. Those elements always form a normal subgroup, straight from the table.
Dot each row of the table into your character to count the pieces, and read normal subgroups from the kernels.
2 · Watch
Take it off screen
Where it sits
This opens up
Nothing builds on it yet.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.