Characters and the Orthogonality Relations
Reduce a representation to the trace of each group element, and prove the resulting functions are orthogonal for an inner product on class functions.
What a learner can do afterwards
- Show a character is constant on conjugacy classes and independent of the chosen basis
- Prove the first orthogonality relation for irreducible characters
- Count the irreducible representations against the number of conjugacy classes
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A representation gives each group element a square matrix, so the group can act by stretching and turning space. The character of the representation at that element is the trace of its matrix. The trace is a single number: add the entries down the main diagonal.
A new basis replaces every matrix by a conjugate, which pictures the same stretch drawn on a new grid. The trace survives conjugation, so the character never sees which basis you chose. Conjugate group elements get conjugate matrices, so they share one character value: the character is constant on each conjugacy class.
Take class functions with an inner product that averages over the group. Two irreducible characters have inner product 1 when they are the same and 0 when they differ. That is the first orthogonality relation: distinct irreducible characters sit at right angles to each other.
When you finish, count. The number of irreducible representations equals the number of conjugacy classes, so a missing character means the table is incomplete.
The trace of each matrix gives a class function, and distinct irreducible characters are orthogonal under the group average.
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Where it sits
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.