The Module Structure of a Group Action on a Polynomial Ring

✍ Scribed by Dikran B Karagueuzian; Peter Symonds

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 141 KB
• Volume: 218
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.7867

✦ Synopsis

For any representation of a p-group G on a vector space of dimension 3 over a finite field k of characteristic p, we show how the symmetric algebra, regarded as a kG-module, can be expressed as a direct sum of kG-modules, each one of which is isomorphic to a summand in low degree. It follows that, for any group G, only a finite number of isomorphism classes of summands can occur. ᮊ 1999 Academic Press 1. INTRODUCTION

1.1. Results

Let k be a finite field of q s p l elements, Let G be a p-group, and let M be a kG-module of dimension 3. We denote by S the symmetric algebra on M. This is, of course, equivalent to letting S be the polynomial ring in w x three variables, S s k x, y, z , and stating that G acts by graded ring automorphisms over k, and that the action on the homogeneous part of degree 1 is isomorphic to M. We are concerned with describing S as explicitly as possible as a kG-module.