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The regular representation of $C_{2}$ over a field of characteristic not $2$ is the direct sum of the trivial and sign representations

Example

Let $C_{2} = {e, s}$ with $s^{2} = e$ , and let $k$ be a field of characteristic not $2$ . The regular representation of $C_{2}$ on $k [C_{2}]$ splits as the direct sum of the trivial line and the line on which $s$ acts by $- 1$ ; after identifying $C_{2} ≅ S_{2}$ by $s \mapsto (12)$ , this second line is the sign representation.

Facts & Assumptions

Given: A field $k$ with $char k \neq 2$ and the regular representation of $C_{2}$ .

[L1]

In the regular representation, $s \cdot [e] = [s]$ and $s \cdot [s] = [e]$ (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

[L2]

The trivial representation has $s$ acting by $+ 1$ , and under the identification $C_{2} ≅ S_{2}$ the sign representation has $s$ acting by $- 1$ (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets, The sign representation of $S_{n}$ and the restriction ${Res}_{H}^{G} (V)$ of a representation to a subgroup).

Verification

technique · direct

1.1

Put $u : = [e] + [s]$ and $v : = [e] - [s]$ . By [L1], $s \cdot u = [s] + [e] = u$ and $s \cdot v = [s] - [e] = - v$ .

L1givenalgebra

2.1

Because $char k \neq 2$ , the two vectors $u$ and $v$ are linearly independent and span $k [C_{2}]$ : one has $[e] = \frac{1}{2} (u + v)$ and $[s] = \frac{1}{2} (u - v)$ . Therefore $k u$ is the trivial line of [L2], $k v$ is the sign line of [L2], and the regular representation is their direct sum.

step 1.1L2givenalgebra∎

Depends on

Used by

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Sources

Peter Webb, A Course in Finite Group Representation Theory, Example 1.1.6 (standard reference, not scraped)

The regular representation of C2C2​ over a field of characteristic not 22 is the direct sum of the trivial and sign representations