Why should bosons be in the adjoint representation of the gauge group? Is there a deep mathematical reason for why bosons should be in the adjoint representation of the gauge group rather than any other representation?
 A: I guess by "bosons" you're referring to gauge bosons?
If so then start with some matter field  $ \psi(x)$  which transforms under the gauge group.  For local gauge transformations the gauge group element $g$ is spacetime dependent $g(x)$, and the transformation is
$$\psi(x) \longrightarrow \psi'(x) = g(x)\psi(x).$$
Derivatives would transform as
$$\partial_{\mu}\psi(x) \longrightarrow g(x)\partial_{\mu}\psi(x)+(\partial_{\mu}g(x))\psi(x),$$
i.e. inhomogeneously.  We would like a gauge covariant derivative $D_{\mu}$ which transforms homogeneously as
$$D_{\mu}\psi(x) \longrightarrow g(x)D_{\mu}\psi(x).$$
To achieve this, we define
$$D_{\mu}\psi = \partial_{\mu}\psi - A_{\mu}\psi,$$
where $A_{\mu} = \mathbf{A}_{\mu} \cdot\boldsymbol{{\tau}}$ and $\boldsymbol{\tau}$ are the generators of the Lie algebra of the gauge group and $A_{\mu}$ is our bosonic gauge field.  This introduction of gauge bosons via the derivative term is sometimes referred to as minimal coupling.
In order to achieve this, $A_{\mu}$ is forced to have the transformation law
$$A_{\mu} \longrightarrow A'_{\mu} = gA_{\mu}g^{-1} + (\partial_{\mu}g)g^{-1}.$$
Just looking at how the $A_{\mu}$ are transforming under the group action (the first term), we recognize the adjoint representation.
Of course, on the global stage, the fields $\psi$ can be interpreted as bundle sections and the gauge fields as bundle connections.  $A_\mu$'s transformation law will be recognisable as a transformation of connection coefficients under the action of the bundle's structure group.  A good reference is Nakahara, or this link.
