Characters and the Orthogonality Relations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Traces that tell representations apart

Mathematics · Abstract Algebra · ages 23-24
Name ______________________   Date ____________

  1. Two elements are conjugate in the group. What must their character values do?

    • Agree with each other
    • Differ by a sign
    • Multiply to give 1

  2. In S3, which pair of elements must take the same value under every character?

    • (12) and (123)
    • The transpositions (12) and (13)
    • The identity and (123)
    • The identity and (12)

  3. Two matrices represent the same group element in different bases. A student says their traces must match. Is that right?

    Circle one:   True   False

  4. What do you add to get the character of a group element?

    • The entries across the first row
    • The entries down the main diagonal
    • The entries down the first column

  5. Mia recomputes characters in a new basis and gets the same numbers. Why?

    • Trace survives conjugation, so the basis cannot change the values
    • The group grew smaller when she changed the basis
    • Conjugacy classes merge under a new basis

  6. The standard representation of S3 is 2-dimensional. What is its character value at the identity?

    Answer: ______________

  7. A student says S3 has exactly 3 irreducible representations because it has 3 conjugacy classes. Is that right?

    Circle one:   True   False

  8. The standard character of S3 takes values 2, 0, -1 on its three conjugacy classes. What is the sum of the squares of these values?

    • 1
    • 6
    • 5
    • 3

  9. Leo says two non-isomorphic irreducible representations have inner product 1. What is wrong?

    • Nothing is wrong with his claim
    • Inner products refuse to compare two characters
    • Different irreducibles are orthogonal, so the value is 0

  10. The conjugacy classes of a group are C1, C2, C3, C4, C5 and C6. Type the number of its irreducible representations.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Traces that tell representations apart W1-mt_RleR7YGXjX-s1

  1. Agree with each other · Characters are constant on conjugacy classes, so the values agree.
  2. The transpositions (12) and (13) · Conjugate elements share character values, and the transpositions (12) and (13) are conjugate in S3.
  3. True · Trace is cyclic, so similar matrices share it and characters ignore the chosen basis.
  4. The entries down the main diagonal · The character is the trace, which adds the main diagonal.
  5. Trace survives conjugation, so the basis cannot change the values · A basis change only conjugates matrices, which the trace ignores.
  6. 2 · The character at the identity is the trace of the identity matrix, which is the dimension 2.
  7. True · S3 has three conjugacy classes, and the count of irreducibles always matches that number.
  8. 5 · Squaring gives 4 + 0 + 1 = 5.
  9. Different irreducibles are orthogonal, so the value is 0 · Non-isomorphic means different, and different irreducibles give 0.
  10. 6 · Six classes carry six irreducible characters.
Worksheet · LightMySky