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- R. James Shank
- 1998

We use the theory of S.A.G.B.I. bases to 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 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… (More)

We study the transfer homomorphism in modular invariant theory paying particular attention to the image of the transfer which is a proper non-zero ideal in the ring of invariants. We prove that, for a p-group over Fp whose ring of invariants is a polynomial algebra, the image of the transfer is a principal ideal. We compute the image of the transfer for… (More)

It is well-known that the ring of invariants associated to a non-modular representation of a nite group is Cohen{Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen{Macaulay and computing the depth is often very diicult. In this paper 1 we obtain a simple… (More)

- Peter Fleis hmann, Gregor Kemper, R. James Shank
- 2002

Let W be a finite-dimensional Z/p-module over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W ] Z/p , is called the Noether number of the representation, and is denoted by β(W). A lower bound for β(W) is derived, and it is shown that if U is a Z/p submodule of W , then β(U) 6 β(W). A set of… (More)

- R. James Shank, David L. Wehlau
- J. Symb. Comput.
- 2002

Let V be a finite dimensional representation of a p-group, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V ] G , has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V ] G. We use these methods to analyse k[2V 3 ] U 3 where U… (More)

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants of m copies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré… (More)

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… (More)

- H E A Campbell, R J Shank
- 2009

In this paper, we study the vector invariants, F[m V 2 ] Cp , of the 2-dimensional indecomposable representation V 2 of the cylic group, C p , of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman [18] who showed that this ring required a generator of degree m(p − 1), thus demonstrating that the result of… (More)

There is a relationship between the covariants of binary forms, a central topic in classical invariant theory, and the invariants of modular representations of cyclic groups of prime order. This relationship was identified by Gert Almkvist [1] and used implicitly in both [15] and [17]. In this note we investigate the relationship and provide a progress… (More)