Formal Modular Seminvariants

We construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We then observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p − 3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p− 2. Introduction Let k be a field and define a linear function σ by σ(x1) = x1 and, for i > 1, σ(xi) = xi + xi−1. Extend σ to an algebra automorphism of the polynomial algebra k[x1, . . ., xn]. If f ∈ k[x1, . . ., xn] and σ(f) = f , then f will be called σ-invariant. Since σ is a degree preserving map, any σ-invariant polynomial is a sum of homogeneous σ-invariant polynomials. Thus we will restrict our attention to homogeneous polynomials. Let k[x1, . . ., xn] σ denote the ring of σ-invariant polynomials. Suppose that p is a prime number and let Fp denote the field with p elements. If k = Fp and n ≤ p, then σ generates a group isomorphic to Z/p and we denote k[x1, . . ., xn] by Fp[x1, . . ., xn] . The action of Z/p induced by σ on the degree one polynomials of Fp[x1, . . ., xn] is the indecomposable modular representation of dimension n. The study of Fp[x1, . . ., xn] Z/p has a long history going back at least to L. E. Dickson’s Madison Colloquium [5]. From Dickson’s perspective the problem is an extension of classical invariant theory and the elements of Fp[x1, . . ., xn] Z/p are the formal modular semivariants of a binary (n − 1)–form [5, III]. Dickson gave a complete description of Fp[x1, . . ., xn] Z/p for n = 2 and n = 3. He gave a generating set for n = 4, p = 5. G. Almkvist, in [1], described the set of relations for n = 4, p = 5. W. L. G. Williams, in [10], constructed a generating set for n = 4, p = 7. The primary purpose of this paper is to describe a generating set for n = 4 and n = 5 for all p ≥ 5. If the characteristic of k is zero, then σ generates a group isomorphic to Z. In this case we denote k[x1, . . ., xn] σ by k[x1, . . ., xn] . Let Q denote the rational num1991 Mathematics Subject Classification. 13A50.