Let $X$ be a path-connected manifold nice enough such it's universal covering
space $p:\widetilde{X} \to X$ exists, $k$ a field. Then there exist a wellknown
correspondence
$$
\{\textit{linear}\text{ representations of }\pi_1(x,x)\} \leftrightarrow
\{\text{local systems of }\textit{vector spaces}\text{ on }X\}
$$
between $k$ linear finite dimensional representations of a fundamental group
$\pi_1(X,x)$ and local systems of $k$ vector spaces.
The map in one direction is defined as follows:
Take a $k$ linear rep $\rho: \pi_1(X,x) \longrightarrow \operatorname{GL}(V) $
where $V$ is a $k$ space and consider the associated $V$-bundle as
quotient space
$\widetilde{X}\times_{\rho} V :=(\widetilde{X}\times V)/\pi_1(X,x) $
where $\pi_1(X,x)$ acts on $\widetilde{X}\times V$ via
$$ g \cdot(x,v) := (g \cdot x, \rho(g)\cdot v ) $$
where $g$ acts at the left via monodromy on the covering space.
Obviously the projection to the first coordinate
$p:\widetilde{X}\times_{\rho} V \to X$ has fiber $V$ and if we endow
$V$ with the discrete topology we obtain a local system $\mathcal{F}_{\rho}$
on $X$ defined by sections
$$\mathcal{F}_{\rho}(U)= \{s:U \to p^{-1}(U) \ \vert p \cdot s =1_U \} $$
for open $U \subset X$. It's easy to check that if $U $ is contractible,
then $p^{-1}(U)\cong U \times V$ and since $V$ has discrete topology,
$\mathcal{F}_{\rho}(U) \cong V$, so it's a local system.
Question: Is there an explicit construction known to go in another
direction? To start with an local system $\mathcal{F}$ with fibre $V$ and
construct from it explicitly a representation $\rho_F: \pi_1(X,x) \longrightarrow \operatorname{GL}(V) $?
I know that it's rather easy to construct it abstractly:
Let $g=[\gamma] \in \pi_1(X,x)$ be a class of a loop, then since $[0,1]$ is
contractible, all local systems on $[0,1]$ are constant sheaves, therefore we
have a chain of abstract isomorphisms
$$ \gamma^*\mathcal{F}_0 \cong \gamma^*\mathcal{F}([0,1])\cong \gamma^*\mathcal{F}_1 =V.$$
Can this isomorphism of $V$ be written down in explicit terms as an element of
$\operatorname{GL}(V)$ if we pick a basis $e_1,\dotsc, e_n$ of $V \cong k^n$?
Motivation of the question: In Geordie Williamson's An illustrated guide to perverse sheaves
in example 5.11 one considers for $X:= \mathbb{C}^*$ and $k:=\mathbb{C}$ the
covering map $f:\mathbb{C}^* \to \mathbb{C}^*: z \mapsto z^m$.
Let $\underline{k}$ be the constant sheaf on $\mathbb{C}^*$ with value
$k=\mathbb{C}$ regarded as 1D vector space.
One considers the pushforward sheaf $f_*\underline{k} $ which has as stalk
at $x=1$ the functions from the $m$-set $f^{-1}(x)$ to $k$, which is isomorphic to $k^m$.
And then it is claimed that $f_*\underline{k} $ is a local system
determined by the action of the monodromy on the $m$-th roots of $1$.
And I was wondering how to check this claim explicitly, even though this sounds
plausible. To come back to the question I posed above it suffices to check that
$f_*\underline{k} $ induces the repr $\pi_1(\mathbb{C}^*,1) \cong \mathbb{Z} \to
\operatorname{GL}_m(\mathbb{C})$ which maps the generator $1$ to
$m$-cycle mapping for a fixed ordered basis $e_1,e_2,\dotsc, e_m$ of $k^m$ the basis vector
$e_i$ to $e_{i+1}$.
Best Answer
Here is a way to fill in the details. For simplicity, write $I = [0,1]$ for the interval, and $\exp \colon I \to S^1$ for the function $x \mapsto e^{2\pi i x}$. So let $\gamma = \exp \colon I \to S^1$ be a generator of $\pi_1(S^1,1)$. To get the map $\gamma_* \colon f_*\underline k \to f_*\underline k$, we need to find a trivialisation of $\gamma^*f_* \underline k$. Luckily, this is not too hard. First, consider the pullback square $$\begin{array}{ccc}X & \stackrel p \to & S^1 \\ \!\!\!\!{\small q}\downarrow & & \downarrow{\small f}\!\!\! \\ I & \stackrel\gamma\to & S^1.\!\end{array}$$ Then $X$ is a disjoint union $\coprod_{a \in \mathbf Z/m\mathbf Z} I$ of $m$ copies of $I$ that each map isomorphically to $I$ under $q$, and $p \colon X \to S^1$ is given on the $a$-th copy by $x \mapsto \exp((x+a)/m)$. In other words, $X$ is obtained from the top copy of $S^1$ by breaking it up into $m$ pieces by cutting at the roots of unity.
Now we claim that $\gamma^*f_*\underline k$ is naturally isomorphic to $q_*p^*\underline k$. There is always a map $$\gamma^*f_*\underline k \to \gamma^*f_*p_*p^* \underline k = \gamma^*\gamma_*q_*p^*\underline k \to q_*p^*\underline k$$ coming from the unit $1 \to p_*p^*$ and counit $\gamma^*\gamma_* \to 1$ of the adjunctions $p^* \dashv p_*$ and $\gamma^* \dashv \gamma_*$ respectively. In this case, the map $\gamma^*f_* \underline k \to q_*p^*\underline k$ is an isomorphism, either by checking directly at stalks, or by invoking (an easy case of) the proper base change theorem.
Of course $p^*\underline k$ is just the constant sheaf $\underline k$ (this holds for pullback of any constant sheaf along any continuous map), and $q_*\underline k$ is isomorphic to $\bigoplus_{a \in \mathbf Z/m\mathbf Z}\underline k$, giving the required trivialisation. Under this isomorphism, the fibre at $0 \in I$ is $\bigoplus_{a \in \mathbf Z/m\mathbf Z} \underline k_{\exp(a/m)}$, whereas the fibre at $1 \in I$ is $\bigoplus_{a \in \mathbf Z/m\mathbf Z} \underline k_{\exp((a+1)/m)}$ (where we express everything in terms of the stalks of $\underline k$ in the upper copy of $S^1$). Both of these are isomorphic to $(f_*\underline k)_1$, but in ways that differ by a cyclic shift. $\square$
Summary. Don't just compute the isomorphism types of various vector spaces, but remember the isomorphisms. All the data is there, but only once you choose a trivialisation of $\gamma^* \mathscr F$.
Remark. There are alternative viewpoints that are also fruitful. For instance, locally trivialising $\mathscr F$ and comparing the trivialisations on the intersections gives a cocycle in $H^1(X,\operatorname{GL}_r(\underline k))$ (cohomology in the constant sheaf of non-abelian groups $\operatorname{GL}_r(\underline k)$). This set also classifies $\operatorname{GL}_r(k)$-torsors over $X$, which is the same thing as $\operatorname{GL}_r(k)$-covering spaces. These are again in correspondence with $\operatorname{Hom}(\pi_1(X,x),\operatorname{GL}_r(k))$. But we have gained a point of view using local trivialisations of $\mathscr F$ on $X$, which is often easier to work with than global trivialisations of $\gamma^* \mathscr F$ on $I$.