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.
More to read
11.5 Matrices and Matrix Operations
Matrices act on vectors by stretching, rotating, and shearing, and the trace boils that action down to one number. Adding the diagonal entries sounds arbitrary, but cyclicity makes it basis free. Characters exploit exactly this: the trace of a representing matrix cannot see which basis was chosen. Matrix operations are therefore the computational engine under every character value.
Visualizing linear transformations
A linear transformation is fixed by what it does to basis vectors, and changing the basis conjugates its matrix. Trace survives conjugation, which is why characters are well defined on group elements rather than on matrices. Visualizing maps as deformed grids makes conjugation concrete: same stretch, new grid lines. That picture is the intuition behind class constancy.
7.5 Matrices and Matrix Operations
Adding and multiplying matrices follows rules that mirror the underlying maps: multiplication composes actions, and the trace extracts their fixed content. Powers and products of matrices stay inside the same arithmetic. Character computations lean on all of it, from multiplying representing matrices to summing diagonal entries. Fluency here pays off the moment class sums appear.
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Where it sits
18 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.