---- Let $V$ be a $1$-[[dimension|dimensional]] [[vector space]] over a [[field]] $\mathbb{F}$ (often $\mathbb{F}=\mathbb{C}$) so that $\text{GL}(V) \cong \text{GL}_{1}(\mathbb{C})\cong \mathbb{C}^{\times}$[[unit|.]] > [!definition] Definition. ([[sign representation]]) > The **sign representation** of the [[symmetric group]] $S_{n}$ is characterized by the [[group homomorphism|homomorphism]] $\begin{align} \Sigma: S_{n} \to \mathbb{C}^{\times} \\ \sigma \mapsto \text{sign}(\sigma) \end{align}$ >where $\text{sign}(\sigma)$ is the [[parity of a permutation|sign of the permutation]] $\sigma \in S_{n}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```