Symmetric and alternating-squares of linear representation

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Definition

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

$S^2\rho (g)(v\otimes w)=\rho (g)(v)\otimes \rho (g)w$ for $v\otimes w\in S^{2}V$, ${\mathrm{\Lambda }}^2\rho (g)(v\otimes w)=\rho (g)(v)\otimes \rho (g)w$ for $v\otimes w\in {\mathrm{\Lambda }}^{2}V$.

Facts

• Even if $\rho$ is irreducible, $S^2\rho$, ${\mathrm{\Lambda }}^2\rho$ need not be irreducible. See the example given in this article.

Characters of the symmetric and alternating-squares

For a representation $\rho$, write ${\chi }_{\rho }$ for its character.

Then

${\chi }_{S^2\rho }=\frac{1}{2}({\chi }_{\rho }(g{)}^2+{\chi }_{\rho }(g^2))$, and

${\chi }_{{\mathrm{\Lambda }}^2\rho }=\frac{1}{2}({\chi }_{\rho }(g{)}^2-{\chi }_{\rho }(g^2))$.

Example

Dihedral group of order 8

Further information: linear representation theory of dihedral group:D8

Consider the dihedral group:D8 $⟨x,a|a^4=x^2=e,xa{x}^{-1}=a^{-1}⟩$ which has a two-dimensional faithful irreducible representation given by

$\rho (a)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\rho (x)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$.

Calculating $S^2\rho$ gives a non-trivial three-dimensional representation of the group:

$S^2\rho (a)=\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 0\\ 1& 0& 0\end{array}\right),S^2\rho (x)=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right)$,

which is reducible.

Calculating ${\mathrm{\Lambda }}^2\rho$ gives a non-trivial one-dimensional representation of the group:

${\mathrm{\Lambda }}^2\rho (a)=\left(\begin{array}{c}1\end{array}\right),{\mathrm{\Lambda }}^2\rho (x)=\left(\begin{array}{c}-1\end{array}\right)$.