invariant theory - Generalized symmetric algebras and Dickson algebras over ${mathbb F}_p$.

Four quick references that contain substantial info on your questions (for more, it'd be good to know what exactly you would like to know...):

de Concini, C.; Procesi, C.
A characteristic free approach to invariant theory.
Advances in Math. 21 (1976), no. 3, 330--354.

Grosshans, F. D.
Vector invariants in arbitrary characteristic.
Transform. Groups 12 (2007), no. 3, 499--514.

Stepanov, S. A.
Vector invariants of symmetric groups in the case of a field of prime characteristic. Discrete Math. Appl. 10 (2000), no. 5, 455--468

Stepanov, Serguei A.
Orbit sums and modular vector invariants. Diophantine approximation, 381--412,
Dev. Math., 16, Vienna, 2008.