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The cyclic fixed-space data recovers an $S_{3}$ rational character

Example

Let $1$ , $sgn$ , and $χ_{s t d}$ be the usual rational characters of $S_{3}$ . Their fixed-space dimensions on the cyclic subgroups $1$ , $C_{2}$ , and $A_{3}$ are

$\begin{array}{cccc} 1 & C_{2} & A_{3} \\ 1 & 1 & 1 & 1 \\ sgn & 1 & 0 & 1 \\ χ_{s t d} & 2 & 1 & 0 \end{array} .$

Hence a character $a \cdot 1 + b \cdot sgn + c \cdot χ_{s t d}$ is recovered uniquely from its cyclic fixed-space data.

Facts & Assumptions

Given: The group $S_{3}$ , its cyclic subgroups $1$ , $C_{2}$ , and $A_{3}$ , and a rational character $x = a \cdot 1 + b \cdot sgn + c \cdot χ_{s t d}$ .

[F1]

Cyclic fixed-space data determines a rational virtual character (Cyclic fixed-space dimensions detect rational virtual characters).

[F2]

For a representation $V$ , the fixed subspace $V^{H}$ is the subspace of vectors fixed by every element of $H$ (The fixed subspace $V^{G}$ of a representation).

[A1]

On $S_{3}$ , the one-dimensional sign representation acts trivially on $A_{3}$ and by $- 1$ on a transposition, while the two-dimensional standard representation is fixed pointwise by the identity, has a one-dimensional fixed line for a transposition, and has no nonzero fixed vector for a $3$ -cycle.

Verification

technique · direct

1.1

By [F2] and [A1], the trivial representation has fixed-space dimensions $(1, 1, 1)$ on $(1, C_{2}, A_{3})$ , the sign representation has $(1, 0, 1)$ , and the standard representation has $(2, 1, 0)$ . This is exactly the displayed table.

F2A1givenalgebra

2.1

Therefore the cyclic fixed-space data of $x = a \cdot 1 + b \cdot sgn + c \cdot χ_{s t d}$ is $(a + b + 2 c, a + c, a + b)$ . The coefficient matrix $(\begin{array}{ccc} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array})$ has determinant $2 \neq 0$ , so these three numbers determine $a$ , $b$ , and $c$ uniquely.

step 1.1algebra

3.1

Thus the cyclic fixed-space data recovers the rational character $x$ , which is the concrete $S_{3}$ instance of [F1].

F1step 2.1algebra∎

Depends on

Used by

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Sources

Tammo tom Dieck, Representation Theory, Theorem (4.5.3) (standard reference, not scraped)
Tammo tom Dieck, Representation Theory, Section 4.4 Example (4.4.5) (standard reference, not scraped)

The cyclic fixed-space data recovers an S3S3​ rational character