Representation

Definition Let $G$ be a finite group and $V$ a finite dimensional complex vector space. A representation of $G$ on $V$ , or a $G$-representation, is a homomorphism $\rho :G\to GL(V)$. We denote such a representation by $(\rho ,V)$, or sometimes just by $\rho$ or $V$. Upon choosing a basis $B$ of $V$, for $V$an n-dimensional space, we get an isomorphism $GL(V)\cong G{L}_n(C)$, $T\to [T{]}_B^B$

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Permutation representation

Let $G$ be a finite group and $X$ a finite set. If $G$ acts on $X$, then it induces a representation of $G$ on $\mathbb{C}(X)$ , the space of complex valued functions on $X$ by $(\phi (g)f)x=f(g^{-1}x).$Indeed, we check that $\phi (gh)=\phi (g)\phi (h):(\phi (gh)f)x=f(g{h}^{-1}x)=f(h^{-1}{g}^{-1}x)=(\phi (g)\phi (h)f)x$ The representation $\mathbb{C}(X)$ is called the permutation representation associated with the action. Remark The Character ${\chi }_{\phi }(g)$ is the number of fixed point of $g$ in $X$.

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Regular representation The regular representation of a group $G$ is the permutation representation of $G$ associated with its action on itself $x\to gx.$ Remark The character of regular representation is ${\chi }_{\phi }(g)=\{\begin{array}{ll}|G|,& \text{if}g=e\\ 0,& \text{otherwise}.\end{array}$

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Dual representation The dual representation $({\rho }^{\ast },V^{\ast })$ of a representation $(\rho ,V)$ group $G$ is defined by ${\rho }^{\ast }(g)(f)=\rho (g^{-1}{)}^T(f)$