Line bundles, line bundles on a homogeneous space, and sections of line bundles I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and  sections of line bundles. These concepts are on page 140 (the first paragraph of section 9.3) of the book Young Tableaux by Fulton. 
I know that $R \times S^{1}$, where $S^{1}$ is the unit circle and $R$ is the real line, is a line bundle. Möbius strip is also a line bundle. Are there other specific examples of line bundles? 
How to understand the sentence in the first paragraph of section 9.3 (page 140) in the book of Fulton clearly: There is a general procedure for producing representations as sections of a line bundle on a homogeneous space?
I also have a question in the first paragraph of the book Young Tableaux on page 131. It is said that if $F$ is a subspace of $E$ of co-dimension $d$, then the kernel of the map from $\wedge^{d}(E)$ to $\wedge^{d}(E/F)$ is a hyperplane in $\wedge^{d}(E)$; assigning this hyperplane to $F$ gives a mapping $Gr^{d}E \to P^{*}(\wedge^{d}E)$. What are the maps $\wedge^{d}(E) \to \wedge^{d}(E/F)$ and $Gr^{d}E \to P^{*}(\wedge^{d}E)$ explicitly?
Thank you very much.
 A: The formation of exterior powers is a functor, so the quotient map
$E \to E/F$ induces a map $\wedge^d E \to \wedge^d(E/F)$.  It is not hard to
see that this latter map is surjective.  Since $E/F$ is of dimension $d$ by assumption, its top exterior power $\wedge^d(E/F)$ is one-dimensional, and so
the kernel of $\wedge^d E \to \wedge^d(E/F)$ is of codimension one, i.e. is
a hyperplane.
Now given $F$, this kernel is a hyperplane in $\wedge^d E,$ and so 
we get a map $Gr^dE \to P^*(\wedge^d E)$ sending each element $F$ of the source
(i.e. each codimension $d$ subspace $F$ of $E$) to the kernel of the associated
map $\wedge^d E \to \wedge^d (E/F)$, which as we have just seen is a hyperplane
in $\wedge^d E$, i.e. is an element of $P^*(\wedge^d E)$.  
As for your question about line bundles, you might want to learn more basic theory before trying to understand the statement about sections of homogeneous line bundles.  For example, the examples you wrote down are line bundles in the smooth category, whereas Fulton is talking about line bundles, and sections thereof, in the algebraic category.
One place to start would be by learning about the tautological bundle over a projective space, since this is an example of the kind of homogeneous line bundle that Fulton is talking about.
