Confusion on induced representations Let $H$ be a subgroup of $G$. Let $R = \{\sigma_1, \ldots, \sigma_n\}$ be a set of representatives of $G/H$. Let $\theta: H\to\text{GL}(W)$ be a representation of $H$. Given this information, we can construct an induced representation $\rho: G\to\text{GL}(V)$, where
$$V = \bigoplus_{i=1}^n W_{\sigma_i}$$.
This is now where I get confused. We define $W_{\sigma_i}$ as $W_{\sigma_i} = \{\sigma_i\cdot w: w\in W\}$, but $\sigma_i$ is generally an element of $G$, not $H$. How can we know how elements of $G$ act on $W$, if $\theta$ is defined only on $H$.
If we know how $G$ acts on $W$, then is $\rho$ just defined as follows:
$$\rho_g(w_{\sigma_1}\oplus\cdots\oplus w_{\sigma_n}) = g\cdot w_{\sigma_1}\oplus\cdots\oplus g\cdot w_{\sigma_n}?$$
 A: The key idea is the permutation representation $G\curvearrowright G/H$. The fact $R$ is a left transversal means if we multiply any coset representative $\sigma\in R$ by any group element $g\in G$, we can "slide" the $g$ across the $\sigma$, in the sense that we can equate $g\sigma$ with $\sigma'g'$ for some other representative $\sigma'\in R$ and element $g'\in G$.
Thus, if you define elements of $\sigma W$ to just be "formal products" of the form $\sigma w$, then $\sigma w$ does not simplify to anything else (in particular, $G$ does not act on $W$), and if we apply $g$ to an element $\sigma w\in\sigma W$, we get $g\sigma w$, which expect/want to be the same as $\sigma'g'w$, which then makes sense because $g'w\in W$ so $\sigma'(g'w)\in\sigma' W$. Thus, there are a bunch of vector subspaces of $V$ parametrized by cosets of $H$, and elements of $G$ permute these subspaces under multiplication just as it permutes the cosets themselves under multiplication. That is, $g(\sigma W)=\sigma'W$ iff $g\sigma H=\sigma'H$.
The simplest example of this is inducing the 1D trivial rep of the symmetric group $S_n$ from the trivial subgroup. The induced representation is just $\mathbb{C}^n$ (or whichever field you want) with permutations acting by permutation matrices! Then $W=\mathbb{C}\times\{0\}^{n-1}$ is just the "$x$-axis," and the collection $\{\sigma W\mid \sigma\in S_n/1\}$ is just the set of all coordinate axes!
In general, if $\rho_W$ is a representation of $H$, then we can extend a basis for $W$ to one for the induced representation $V$, and all $\rho_V(g)$s are expressible as block matrices. (The blocks will all be the same if $G$ is a direct product of $H$ and another subgroup, but generally not otherwise. The blocks in a given matrix will be conjugate elements of $\rho_W(H)$ though.)
Note you can do the same thing with sets in place of vector spaces and group actions in place of representations. If $H$ acts on a set $\Omega$, then we can define the "induced $G$-set" to be the quotient $G\times_H\Omega$, i.e. thet set $G\times\Omega$ mod the equivalence relation $(gh,\omega)\sim(g,h\omega)$. This carries a left action of $G$ via $g(g',\omega)=(gg',\omega)$, which respects the relation. If $R$ is a left transversal for $G/H$, then $G\times_H\Omega$ may be partitioned into subsets $\{\sigma\}\times\Omega$ ($\sigma\in R$). Then, if $g\sigma$ simplifies to $\sigma'g'$, we can say $g(\sigma,\omega)=(\sigma',g'\omega)$.
