For an n-dimensional smooth manifold $M$, We can consider the bundle of 1-jets $J^1 M$ to be a $Gl(n,\mathbb{R})$ bundle over $M$. Representations of this group are straightforward. (Note by $J^1$ I mean $J^{1} _{n}$ always)
I am interested in finding representations of higher nonholonomic jet groups. For the second order nonholonomic jet bundle we have $$\tilde{J}^2 M = J^1 J^1 M$$ which is a $J^1 Gl(n,\mathbb{R})$ bundle. How then to find representations of this group (or more generally representations of $J^1 G$ where $G \subset Gl(n,\mathbb{R})$).
In reading The differential Geometry of Frame bundles, Cordero, Dodson, De Leon (pg 19) say that:
Let $FM(M,\pi_{m},Gl(n,\mathbb{R}))$ denote the frame bundle of the differentiable manifold $M$, $$J_{n}^{1}FM(J_{n}^{1}M,\pi_{M}^{1},J_{n}^{1}Gl(n,\mathbb{R}))$$ the induced $J_{n}^{1}Gl(n,\mathbb{R})$-principal bundle, and let $$FJ_{n}^{1}M(J_{n}^{1}M,\pi_{J_{n}^{1}M},Gl(n+n^{2},\mathbb{R}))$$ be the frame bundle of $J_{n}^{1}M$.
Theorem 2.1.2 There exists a canonical injective homomorphism of principal bundles
$$j_{M}:J_{n}^{1}FM\rightarrow FJ_{n}^{1}M$$
over the identity of $J_{n}^{1}M$, with associated Lie group homomorphism
$$j_{n}:J_{n}^{1}Gl(n,\mathbb{R})\rightarrow Gl(n+n^{2},\mathbb{R})$$
Theorem 2.1.3 Let $\pi_{m}:FM\rightarrow M$ and $\pi_{FM}:FFM\rightarrow FM$ be the frame bundle of $M$ and $FM$ respectively. Then, $j_{M}$ induces a bundle homomorphism of $J_{n}^{1}FM|_{FM}$ into $FFM$ with respect to $j_{n}$, i.e.
$$j_{M}(X\cdot Y)=j_{M}(X)\cdot j_{n}(Y)$$
for $X\in J_{n}^{1}FM|_{FM}$ and $Y\in J_{n}^{1}Gl(n,\mathbb{R})$, and the following diagram commutes
$$\begin{array}{ccccc} J_{n}^{1}FM|_{FM} & \rightarrow & j_{M} & \rightarrow & FFM\\ \downarrow & & & & \downarrow\\ \pi_{M}^{1} & & & & \pi_{FM}\\ \downarrow & & & & \downarrow\\ FM & \rightarrow & 1_{FM} & \rightarrow & FM \end{array}$$
Where they note that $FJ^1 M|_{FM}$, the restriction of $F J^1 M$ to $FM$ is canonically isomorphic to $FFM$, the frame bundle of the frame bundle.
Does this embedding of $J^1 Gl(n,\mathbb{R})$ in $Gl(n+n^2 , \mathbb{R})$ then form a representation? I don't quite get the restriction to $FM$ part. Is this homomorphism injective (i.e. is this a faithful representation?)
Anyone who could straighten this out would be much appreciated.