On the Depth of the Invariants of the Symmetric Power Representations of SL2(Fp)

Abstract

We study the depth of the ring of invariants of SL2(Fp) acting on the nth symmetric power of the natural two-dimensional representation for n<p. These symmetric power representations are the irreducible representations of SL2(Fp) over Fp. 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≠7 and n=4, p≠5. However, for n=3, p=7 the depth is 4 and for n=4, p=5 the depth is 5. In these two exceptional cases, the ring of invariants is Cohen–Macaulay.