On the depth of the invariants of the symmetric power representations of SL2(F-p)

We study the depth of the ring of invariants of SL2(F-p) acting on the nth symmetric power of the natural two-dimensional representation for n < p. Th ese symmetric power representations are the irreducible representations of SL2(F-p) over F-p. We prove that, when the greatest common divisor of p - 1 and n is less than or equal to 2, the depth of the ring of invariants is 3 . We also prove that the depth is 3 for n = 3, p not equal 7 and n = 4, p n ot equal 5. However, for n = 3, p = 7 the depth is 4 and for n = 4, p = 5 t he depth is 5. In these two exceptional cases, the ring of invariants is Co hen-Macaulay. (C) 1999 Academic Press.