Decomposing a Representation from Its Character · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Reading a group from its character table

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

  1. To get a multiplicity, you multiply matching entries of two characters and then what?

    • Add them up
    • Divide by the first entry
    • Keep only the largest product

  2. A student says distinct rows of a character table are always orthogonal. Is that right?

    Circle one:   True   False

  3. Which subset of S3 is the kernel of the sign character?

    • All the transpositions
    • The identity plus the two 3-cycles
    • The whole group S3
    • Only the identity

  4. In a character table, what does one row hold?

    • One group element written out in full
    • One conjugacy class listed by its members
    • One irreducible character across all the classes

  5. The regular representation of the 3-element cyclic group contains each irreducible with some multiplicity. What is the multiplicity of a nontrivial one-dimensional character?

    Answer: ______________

  6. In the same decomposition, the sign character is (1, -1, 1). What is its multiplicity?

    • 0
    • 1
    • 2
    • 3

  7. The permutation character (3, 1, 0) of S3 decomposes with trivial multiplicity 1, sign multiplicity 0, and standard multiplicity 1. How many irreducible constituents are there in total?

    Answer: ______________

  8. A row's value equals its dimension exactly on the identity and two other elements. What do those three form?

    • A normal subgroup
    • A single conjugacy class
    • A second table

  9. Dan says the trivial piece appears twice because the first entry of the character is 2. What is his mistake?

    • One entry decides the count on its own
    • Only the full inner product counts the copies
    • Tables cannot detect the trivial piece

  10. Two distinct irreducible characters of a finite group pair to 0 under the class inner product. A student says this always happens. Is that right?

    Circle one:   True   False

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Answer key

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

Reading a group from its character table W1-mt_ccEcWW9xAa-s1

  1. Add them up · The inner product multiplies matches and adds, like a dot product.
  2. True · Rows of a character table are orthonormal under the class-weighted inner product.
  3. The identity plus the two 3-cycles · The kernel collects elements sent to 1, namely the identity plus the two 3-cycles.
  4. One irreducible character across all the classes · Each row is a single irreducible character read across the classes.
  5. 1 · The regular character divided by the group order leaves multiplicity 1 for every irreducible.
  6. 0 · Pairing (3, 1, 0) with the sign character (1, -1, 1) gives (3 - 3 + 0)/6 = 0.
  7. 2 · The trivial multiplicity is 1, the sign multiplicity is 0, and the standard multiplicity is 1, totalling 2 constituents.
  8. A normal subgroup · Elements matching the dimension are the kernel, a normal subgroup.
  9. Only the full inner product counts the copies · Copies come from dotting the whole character, never from one entry.
  10. True · That vanishing is exactly the first orthogonality relation for distinct irreducibles.
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