# Decomposing Symmetric Powers of Certain Modular Representations of Cyclic Groups

@article{Shank2005DecomposingSP, title={Decomposing Symmetric Powers of Certain Modular Representations of Cyclic Groups}, author={R. James Shank and David L. Wehlau}, journal={arXiv: Commutative Algebra}, year={2005}, pages={169-196} }

For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p 2, and show that the Noether number for the representation is p 2 + p−3. We then use the constructed invariants to explicitly describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case. In the final section, we use our…

## 7 Citations

### The invariants of modular indecomposable representations of Z p 2

- Mathematics
- 2008

We consider the invariant ring for an indecomposable representation of a cyclic group of order p 2 over a field F of characteristic p. We describe a set of F-algebra generators of this ring of…

### Rings of invariants for modular representations of the Klein four group

- Mathematics
- 2015

We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation we compute the Noether number and give minimal generating sets…

### The invariants of modular indecomposable representations of $${{\mathbb Z}_{p^2}}$$

- Mathematics
- 2008

We consider the invariant ring for an indecomposable representation of a cyclic group of order p2 over a field $${\mathbb F}$$ of characteristic p. We describe a set of $${\mathbb F}$$-algebra…

### Coinvariants and the regular representation of a cyclic P-group

- Mathematics
- 2013

We consider an indecomposable representation of a cyclic p-group $${Z_{p^r}}$$ over a field of characteristic p. We show that the top degree of the corresponding ring of coinvariants is less than…

## References

SHOWING 1-10 OF 17 REFERENCES

### Symmetric powers of modular representations, hilbert series and degree bounds

- Mathematics
- 2000

Let G = Z p be a cyclic group of prime order p with a representation G → GL(V) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric…

### Depth of modular invariant rings

- Mathematics
- 2000

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For…

### Noether Numbers for Subrepresentations of Cyclic Groups of Prime Order

- Mathematics
- 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…

### S.A.G.B.I. bases for rings of formal modular seminvariants

- Mathematics
- 1998

Abstract. 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…

### The module structure of a group action on a polynomial ring: A finiteness theorem

- Mathematics
- 2007

Consider a group acting on a polynomial ring over a finite field. We study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries. For example,…

### On Cohen-Macaulay rings

- Mathematics
- 1994

In this paper, we use a characterization of R-modules N such that fdRN = pdRN to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting N to be the dth local…

### The Magma Algebra System I: The User Language

- Computer ScienceJ. Symb. Comput.
- 1997

The MAGMA language is presented, the design principles and theoretical background are outlined, and the constructors for structures, maps, and sets are outlined.

### Cohomology of groups

- EngineeringOxford mathematical monographs
- 1991

A rink-type roller skate is provided with a plastic sole plate. To mount a toe stop on the skate, a novel bushing is embedded in the sole plate. The bushing has relatively small diameter ends and a…