Definition: Let $G$ be a group, $\rho : G\rightarrow GL(V)$ and $\rho' : G\rightarrow GL(V')$ be two representations of G. We say that $\rho$ and $\rho'$ are $equivalent$ (or isomorphic) if $\exists \space T:V\rightarrow V'$ linear isomorphism such that $T{\rho_g}={\rho'_g}T\space \forall g\epsilon G$.
But I don't understand why this concept is useful. If two groups $H,H'$ are isomorphic, then we can translate any algebraic property of $H$ into $H'$ via the isomorphism. But I don't see how a property of $\rho$ can be translated to similar property of $\rho'$. Nor I have seen any example in any textbook where this concept is used. Can someone explain its importance?
Best Answer
This is true for all sorts of types of representations. Maybe it is easier to see for permutation representations.
Suppose $G$ is a group that acts on polynomials. It has an element $g$ that swaps $x$ and $y$, and leaves $z$ alone. We could write $g=(x,y)(z)$ if we wanted.
But then some jerk comes along and asks what we'd do if we needed a fourth variable. FINE. $G$ is a group that acts on polynomials. It has an element $g$ that swaps $x_1$ and $x_2$, and leaves $x_3$ alone. We could write $g=(x_1,x_2)(x_3)$ if we wanted.
Nothing important has changed really; we just changed the names of the variables.
We could go further and abbreviate $g=(1,2)(3)$. We replaced the variables with the identifying numbers. Maybe that is convenient. Saves ink (or electrons). No real change though.
Vector space representations are the same. If $G$ acts on polynomials, then I guess we could apply $g$ to $2x + 3y + 5z$ to get $2y + 3x + 5z$. So we have $\rho(g)$ a linear transformation of the vector space with basis $\{x,y,z\}$.
ARGH. I forgot about the jerk. $G$ acts on polynomials, so we could apply $G$ to $2x_1 + 3x_2 + 5x_3$ to get $2x_2 + 3x_1 + 5x_3$. So we have $\rho(g)$ a linear transformation of the vector space with basis $\{x_1,x_2,x_3\}$.
The essential properties of $\rho(g)$ and of $\rho$ are not affected by what labels we give to the basis elements of the vector space. Changing the labels on a basis is exactly and only what $T$ does. $T$ doesn't affect how $\rho(g)$ acts, it just affects where $\rho(g)$ acts.