Symmetric and alternating-squares of linear representation: Difference between revisions

Revision as of 17:55, 12 November 2023

Definition

Let $(ρ, V)$ be a linear representation of a group $G$ . Then we can define the linear representations $(S^{2} ρ, S^{2} V)$ and $(Λ^{2} ρ, Λ^{2} V)$ , the symmetric and alternating-squares of $(ρ, V)$ respectively, by restricting the representation $(ρ \otimes ρ, V \otimes V)$ of $G$ to the eigenspaces corresponding to the symmetric and alternating-squares respectively, that is,

$S^{2} ρ (g) (v \otimes w) = ρ (g) (v) \otimes ρ (g) w$ for $v \otimes w \in S^{2} V$ , $Λ^{2} ρ (g) (v \otimes w) = ρ (g) (v) \otimes ρ (g) w$ for $v \otimes w \in Λ^{2} V$ .

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 ==Definition==
-Let <math>(\rho, V)</math> be a [[linear representation]] of a [[group]] <math>G</math>. Then we can define the linear representations <math>(S^2 \rho, S^2 V)</math> and <math>(\Lambda^2 \rho, \Lambda^2 V)</math>, the '''symmetric and alternating-squares''' of <math>(\rho, V)</math> respectively, by restricting the tensor representation <math>(\rho \otimes \rho, V \otimes V)</math> of <math>G</math> to the eigenspaces corresponding to the symmetric and alternating-squares respectively, that is,
+Let <math>(\rho, V)</math> be a [[linear representation]] of a [[group]] <math>G</math>. Then we can define the linear representations <math>(S^2 \rho, S^2 V)</math> and <math>(\Lambda^2 \rho, \Lambda^2 V)</math>, the '''symmetric and alternating-squares''' of <math>(\rho, V)</math> respectively, by restricting the representation <math>(\rho \otimes \rho, V \otimes V)</math> of <math>G</math> to the eigenspaces corresponding to the symmetric and alternating-squares respectively, that is,
 <math>S^2 \rho(g) (v \otimes w) = \rho(g)(v) \otimes \rho(g) w</math> for <math>v \otimes w \in S^2 V</math>, <math>\Lambda^2 \rho(g) (v \otimes w) = \rho(g)(v) \otimes \rho(g) w</math> for <math>v \otimes w \in \Lambda^2 V</math>.