Let $\mathcal{G}$ be a affine algebraic group scheme(may not be reductive) over a scheme $S$. How to define a rational representaion of $\mathcal{G}$ (over $S$)? Is there always a faithful representation?


Please provide references related to these questions.



Representation of a group scheme
I'm not sure what your current sources are, but the definitions are laid out clearly in SGA3 (by Demazure and Grothendieck) and similarly in the book by Demazure and Gabriel, *Groupes algebriques* (North-Holland, 1970) which was later published in an English translation. (Their designation of this book as "Tome I" is of course unfortunate, since it had no sequel.)


In Demazure-Gabriel, one finds for example an explicit statement about the existence (over a field) of a faithful linear representation, in the affine case: see II, 5.2. This is far into their book but is fairly elementary, just relying on the basic notions.


For a treatment heavily influenced by Demazure-Gabriel (or SGA3), you can also consult the early sections of Jantzen's book *Representations of Algebraic Groups* (Academic Press, 1987; 2nd enlarged edition, Amer. Math. Soc., 2003). See especially I.2 for the notion of rational representation of a group scheme (over any commutative ring).


I should add that Jim Milne has developed a modern textbook version of all this, probably published by now; check his webpage for details: <http://www.jmilne.org/math/>


