This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try.

At the risk of repeating well known stuff I tried to make the question as precise as possible.

Let $\pi: P \to M$ be a $G$-bundle with connection form $\omega \in \Omega^1(P ;\mathfrak{g})$. Let $\rho : G \to Gl(V)$ be a linear representation and $E = P \times_{\rho} V$ the corresponding associated bundle.

Denote by $\Omega^{\bullet}_{\rho}(P ; V) \subset \Omega^{\bullet}(P;V)$ the space of $V$-valued forms satisfying:

- Vertical: $i_X \eta = 0$ for all $X \in VP = \ker \pi_*$.
- $R^*_g \eta = \rho(g^{-1}) \cdot \eta$ for all $g \in G$.

There is a one to one between forms in $\Omega^{\bullet}_{\rho}(P ; V)$ and sections of the bundle $\Omega^{\bullet}(M) \otimes E \to M$.

We have an isomorphism $\varphi : \mathfrak{g} \to VP$ sending $X \to (p \mapsto \frac{d}{dt}(e^{tX}\cdot p))$. Therefore, our connection $\omega \in \Omega^1(P ;\mathfrak{g})$ gives rise to a retraction $\varphi^{-1} \circ \omega :TP \to VP$ and thus to a projection onto the horizontal bundle $h= I-\varphi^{-1} \circ \omega$. This determines a derivation on the complex $\Omega^{\bullet}(P;V)$ called the exterior covariant derivative:

$$D: \eta \mapsto d\eta \circ h$$

This derivation descends to a derivation $\Omega^{\bullet}_{\rho}(P ; V)$.

**Now to the question:**

In the appendix of the book "Dirac operator on riemannian geometry" there's a nice formula for how the exterior covariant derivative acts on $\Omega^{\bullet}_{\rho}(P ; V)$. Let $\eta \in \Omega^r_{\rho}(P,V)$. Here is the formula:

$$D \eta = d\eta + \rho_*(\omega)\wedge \eta$$

Where the wedge of a matrix and vector calued forms is defined as:

$$\rho_*(\omega)\wedge \eta (v_0,...,v_r) = \sum^r_{j=0} (-1)^j \rho_*(\omega(v_j))\cdot \eta(v_0,...,\hat{v_j},...,v_r))$$

First part of question:

How can this formula be neatly derived?It gets messy so quickly for me I don't manage to get very far...

Is there a deeper meaning to wedging a matrix and a vector or is it just a notational conveniance?On the face of it it seems to be a very awkward thing to do.

In the case where $G=GL(V)$ the representation is the identity and $P\cong F(E)$ we can get to a connection on the associated bundle $E$ by simply pulling back along a frame. Indeed if $u: U \to P$ is a local section then the covariant derivative is $\nabla = u^*(\rho_*(\omega)) \in \Omega^1(U ; End(V))$.

This procedure doesn't seem to generalize directly.

How do I get from $\rho_*(\omega)$ to a covariant derivative $\nabla \in \Omega^{\bullet}(M;End(E))$ in the general case?

And finally:

**What's a good source to read about transferring connection information from principal bundles to asscoiated bundles and back?**

I can't stress enough the extent at which i feel this is glossed over in the familiar literature.

anyconnection on a vector bundle, it will be true for a connection induced from a connection on a principal bundle via a representation of the structure group. The key formulas are (3.3.12)-(3.3.14) $\endgroup$4more comments