On the Depth of the Invariants of the Symmetric Power Representations of SL_{2] (F_{p} )

Ž . We study the depth of the ring of invariants of SL F acting on the nth 2 p symmetric power of the natural two-dimensional representation for n p. These Ž . symmetric power representations are the irreducible representations of SL F 2 p over F . We prove that, when the greatest common divisor of p y 1 and n is less p than or equal to 2, the depth of the ring of invariants is 3. We also prove that the depth is 3 for n s 3, p / 7 and n s 4, p / 5. However, for n s 3, p s 7 the depth is 4 and for n s 4, p s 5 the depth is 5. In these two exceptional cases, the ring of invariants is Cohen]Macaulay. Q 1999 Academic Press