# Changes

## Linear representation theory of symmetric group:S3

, 11:44, 18 July 2011
Interpretation as general linear group of degree two
|-
| One-dimensional, factor through the determinant map || a homomorphism $\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast$ || $x \mapsto \alpha(\det x)$ || 1 || 1 || $q - 1$ || 1 || trivial representation
|-
| Unclear || a homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, up to the equivalence $\! \varphi \simeq \varphi^q$, excluding the cases where $\varphi = \varphi^q$ || unclear || $q - 1$ || 1 || $q(q - 1)/2$ || 1 || [[#Sign representation|sign representation]]
|-
| Tensor product of one-dimensional representation and the nontrivial component of permutation representation of $GL_2$ on the projective line over $\mathbb{F}_q$ || a homomorphism $\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast$ || $x \mapsto \alpha(\det x)\nu(x)$ where $\nu$ is the nontrivial component of permutation representation of $GL_2$ on the projective line over $\mathbb{F}_q$ || $q$ || 2 || $q - 1$ || 1 || [[Standard representation of symmetric group:S3|standard representation]]
|-
| Induced from one-dimensional representation of Borel subgroup || $\alpha, \beta$ homomorphisms $\mathbb{F}_q^\ast \to \mathbb{C}^\ast$ with $\alpha \ne \beta$, where $\{ \alpha, \beta \}$ is treated as unordered. || Induced from the following representation of the Borel subgroup: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\beta(d)$ || $q + 1$ || 3 || $(q - 1)(q - 2)/2$ || 0 || --
|-
| Unclear || a homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, up to the equivalence $\! \varphi \simeq \varphi^q$, excluding the cases where $\varphi = \varphi^q$ || unclear || $q - 1$ || 1 || $q(q - 1)/2$ || 1 || [[#Sign representation|sign representation]]
|-
| Total || NA || NA || NA || NA || $q^2 - 1$ || 3 ||