# Changes

## Linear representation theory of symmetric group:S3

, 22:06, 2 July 2011
no edit summary
| 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
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| 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 representationof symmetric group:S3|standard representation]]
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| 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 || --
| [[Sign representation]] || 1 || -3 || 2
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| [[Standard representation of symmetric group:S3|Standard representation]] || 1 || 0 || -1
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| [[Sign representation]] || $\mathbb{Z}$ -- the ring of integers || <matH>\mathbb{Q}[/itex] || $\{ 1,-1 \}$ || gives a representation over any ring; nontrivial for characteristic not equal to $2$
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| [[Standard representation of symmetric group:S3|Standard representation]] || $\mathbb{Z}$ -- the ring of integers || $\mathbb{Q}$ || $\{ 0, 1, -1 \}$ || gives an irreducible representation over any ring of characteristic not equal to $2$
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| Sign || 1 || 0 || 0
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| [[Standard representation of symmetric group:S3|Standard]] || 0 || 1 || 1
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| Sign || 0 || 1
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| [[Standard representation of symmetric group:S3|Standard]] || 1 || 1
|}
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