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 $e_{i+1}$.

`\mathrm`

is not for "`\rm`

in a math-mode environment", but for "math in`\rm`

". For the latter, you want something like`\textrm`

, although actually it's just`\text`

. Note $\text{local systems}$`\text{local systems}`

vs. $\mathrm{local systems}$`\mathrm{local systems}`

. Similarly for`\mathit`

, which in your case should have been`\textit`

. I edited accordingly. $\endgroup$2more comments