Changes

Jump to: navigation, search

Linear representation theory of symmetric group:S3

213 bytes added, 22:06, 2 July 2011
no edit summary
| One-dimensional, factor through the determinant map || a homomorphism <math>\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast</math> || <math>x \mapsto \alpha(\det x)</math> || 1 || 1 || <math>q - 1</math> || 1 || trivial representation
|-
| Tensor product of one-dimensional representation and the nontrivial component of permutation representation of <math>GL_2</math> on the projective line over <math>\mathbb{F}_q</math> || a homomorphism <math>\alpha: \mathbb{F}_q^\ast \to \mathbb{C}^\ast</math> || <math>x \mapsto \alpha(\det x)\nu(x)</math> where <math>\nu</math> is the nontrivial component of permutation representation of <math>GL_2</math> on the projective line over <math>\mathbb{F}_q</math> || <math>q</math> || 2 || <math>q - 1</math> || 1 || [[#Standard representationof symmetric group:S3|standard representation]]
|-
| Induced from one-dimensional representation of Borel subgroup || <math>\alpha, \beta</math> homomorphisms <math>\mathbb{F}_q^\ast \to \mathbb{C}^\ast</math> with <math>\alpha \ne \beta</math>, where <math>\{ \alpha, \beta \}</math> is treated as unordered. || Induced from the following representation of the Borel subgroup: <math>\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\beta(d)</math> || <math>q + 1</math> || 3 || <math>(q - 1)(q - 2)/2</math> || 0 || --
| [[Sign representation]] || 1 || -3 || 2
|-
| [[Standard representation of symmetric group:S3|Standard representation]] || 1 || 0 || -1
|}
| [[Sign representation]] || <math>\mathbb{Z}</math> -- the ring of integers || <matH>\mathbb{Q}</math> || <math>\{ 1,-1 \}</math> || gives a representation over any ring; nontrivial for characteristic not equal to <math>2</math>
|-
| [[Standard representation of symmetric group:S3|Standard representation]] || <math>\mathbb{Z}</math> -- the ring of integers || <math>\mathbb{Q}</math> || <math>\{ 0, 1, -1 \}</math> || gives an irreducible representation over any ring of characteristic not equal to <math>2</math>
|}
| Sign || 1 || 0 || 0
|-
| [[Standard representation of symmetric group:S3|Standard]] || 0 || 1 || 1
|}
| Sign || 0 || 1
|-
| [[Standard representation of symmetric group:S3|Standard]] || 1 || 1
|}
Bureaucrats, emailconfirmed, Administrators
38,910
edits

Navigation menu