Definition of a bundle associated to a character In page 31 of the book "Heat kernels and Dirac operators" we find the following paragraph

The bundle of densities $| \wedge|$ is very closely related to the bundle of volume forms $\wedge^n T^*M;$ the first corresponds to the character $| det(A)|^{-1}$ of $GL(n)$ and the second to the character $det(A)^{-1}$.

What is the definition of a bundle which associated to a character of $GL(n)$ ?
 A: Let $P\to M$ be a principal bundle with structure group $G$.
To any vector space $V$ and representation $\rho \colon G \to GL(V)$, one can define the associated vector bundle $P\times_{\rho}V\to M$, which is a vector bundle over $M$ with fibers isomorphic to $V$.
It is defined as the quotient of $P\times V$ by the diagonal (right-)action of $G$ defined by $(p,v)\cdot g = (p\cdot g, \rho(g^{-1})v)$.
Now, consider the principal frame bundle $F(TM)\to M$.
It is a principal bundle with structure group $GL_n(\Bbb R)$.
Consider the two representations
$$
\rho \colon A \in GL_n(\Bbb R) \longmapsto \det A^{-1} \in \Bbb R^* = GL(\Bbb R),\\
|\rho| \colon A \in GL_n(\Bbb R) \longmapsto |\det A^{-1}| \in \Bbb R^* = GL(\Bbb R).
$$
As one dimensional representations, they are usually called characters.
Then one can show that
$$
\Lambda^n(T^*M) \simeq F(TM)\times_{\rho}\Bbb R \quad \text{and} \quad |\Lambda^n|(T^*M) \simeq F(TM)\times_{|\rho|}\Bbb R.
$$
This is what is meant by "they are the vector bundles associated with the characters $\rho$ and $|\rho|$".
