Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e.

\begin{equation} R_g : M \rightarrow M \qquad g\in G \end{equation}

such that

\begin{equation} [R_g,H]=0\qquad \forall g\in G \end{equation}

Considered an eigenvalue $\epsilon$ of $H$, and the corresponding eigenspace $V$, the $R_g$ are internal to this subspace \begin{equation} R_g(V)\subseteq V\qquad\forall g \end{equation} Therefore, the $R_g$ restricted to $V$ are a representation of $G$. By definition, this representation is irreducible iff it does not exsist a proper $W\subset V$ such that \begin{equation} R_g(W)\subseteq W\qquad\forall g \end{equation}

My question, inspired by the Tinkham's book about group theory, is this: are the two statements

1. The $R_g$ restricted to $V$ are an irreducible representation of $G$

2. Given an element $e\in V$, the space vector generated by the vectors $\{R_g(e):\,\,g\in G\}$ is $V$

equivalent? If yes, how is it demonstrated?

Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e.

\begin{equation} R_g : M \rightarrow M \qquad g\in G \end{equation}

such that

\begin{equation} [R_g,H]=0\qquad \forall g\in G \end{equation}

Considered an eigenvalue $\epsilon$ of $H$, and the corresponding eigenspace $V$, the $R_g$ are internal to this subspace \begin{equation} R_g(V)\subseteq V\qquad\forall g \end{equation} Therefore, the $R_g$ restricted to $V$ are a representation of $G$. By definition, this representation is irreducible iff it does not exsist a proper $W\subset V$ such that \begin{equation} R_g(W)\subseteq W\qquad\forall g \end{equation}

My question, inspired by the Tinkham's book about group theory, is this: are the two statements

1. The $R_g$ restricted to $V$ are an irreducible representation of $G$

2. Given an element $e\in V-\{0\}$, the space vector generated by the vectors $\{R_g(e):\,\,g\in G\}$ is $V$

equivalent? If yes, how is it demonstrated?

p.s. Edited after comments

Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e.

\begin{equation} R_g : M \rightarrow M \qquad g\in G \end{equation}

such that

\begin{equation} [R_g,H]=0\qquad \forall g\in G \end{equation}

Considered an eigenvalue $\epsilon$ of $H$, and the corresponding eigenspace $V$, the $R_g$ are internal to this subspace \begin{equation} R_g(V)\subseteq V\qquad\forall g \end{equation} Therefore, the $R_g$ restricted to $V$ are a representation of $G$. By definition, this representation is irreducible iff it does not exsist a proper $W\subset V$ such that \begin{equation} R_g(W)\subseteq W\qquad\forall g \end{equation}

My question, inspired by the Tinkham's book about group theory, is this: are the two statements

1. The $R_g$ restricted to $V$ is an irreducible representation of $G$

2. Given an element $e\in V$, the space vector generated by the vectors $\{R_g(e):\,\,g\in G\}$ is $V$

equivalent? If yes, how is it demonstrated?

Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e.

\begin{equation} R_g : M \rightarrow M \qquad g\in G \end{equation}

such that

\begin{equation} [R_g,H]=0\qquad \forall g\in G \end{equation}

Considered an eigenvalue $\epsilon$ of $H$, and the corresponding eigenspace $V$, the $R_g$ are internal to this subspace \begin{equation} R_g(V)\subseteq V\qquad\forall g \end{equation} Therefore, the $R_g$ restricted to $V$ are a representation of $G$. By definition, this representation is irreducible iff it does not exsist a proper $W\subset V$ such that \begin{equation} R_g(W)\subseteq W\qquad\forall g \end{equation}

My question, inspired by the Tinkham's book about group theory, is this: are the two statements

1. The $R_g$ restricted to $V$ are an irreducible representation of $G$

2. Given an element $e\in V$, the space vector generated by the vectors $\{R_g(e):\,\,g\in G\}$ is $V$

equivalent? If yes, how is it demonstrated?