Example 16.7.1 (Standard representation of ). The 2-dimensional irreducible of is real: acts on by rotations and reflections, and .
Alternatively: (since is real-valued), and contains no trivial summand, so the -invariant bilinear form must be in . Since contains a trivial summand, an invariant symmetric form exists, confirming is real.
Example 16.7.2 (The quaternion group ). The quaternion group with acts faithfully on by The character takes values and on all other elements—all real. However, this representation is not real: it is quaternionic.
To see this cannot come from a real representation: if acted faithfully on , the invariant inner product would give . But has only 2 elements squaring to (rotations by ), while needs 6 elements () with square . Contradiction.
The quaternionic structure is: view via , and the above matrices correspond to left multiplication by . The antilinear map with is right multiplication by , which commutes with left multiplication.
Exercise 16.7.1 (Exercises).
Classify each irreducible representation of as real, quaternionic, or complex-type. (Hint: all characters of are real-valued, so each is either real or quaternionic. Use the dimension criterion: a quaternionic representation must have even dimension.)
Show that if is a quaternionic representation of (carrying an antilinear with ), then is even.
Prove the Frobenius–Schur indicator formula: for an irreducible representation , (Hint: , and exactly one of these dimensions is 1 when .)
Using the Frobenius–Schur indicator, verify that the 2-dimensional irreducible of is quaternionic.
Show that the representations and of (the 3-dimensional irreducibles with golden-ratio character values) are real representations. (Their characters are real, and they have odd dimension.)
Show solution
(1) All characters of are real-valued (check the character table). A quaternionic representation has with , and pairs basis vectors into orbits of size 2, so must be even. The irreducibles of have dimensions 1, 1, 2, 3, 3. The odd-dimensional ones () are automatically real. For (): compute the Frobenius–Schur indicator . We have with class sizes . Squaring: . So . The indicator is . So is also real.
(2) The antilinear map gives an -linear map whose square is . Viewing as (where ), this gives a real matrix with . The characteristic polynomial has degree and all roots are , which come in conjugate pairs. More directly: makes into an -module, and must be a positive integer.
(3) We have . Using the decomposition and the identities ,, we get the indicator . When , the unique invariant bilinear form is either symmetric (contributing 1 to ) or skew (contributing 1 to ). When , neither nor contains a trivial summand, giving 0.
(4) has 5 conjugacy classes: of sizes 1, 1, 2, 2, 2. For the 2-dimensional rep: . Squaring: ,,,,. So . Indicator: . Quaternionic.
(5) Both and have real characters and odd dimension 3. Since a quaternionic representation must have even complex dimension (exercise 2), odd-dimensional representations with real characters are automatically real.