2013/08/16 by H. E. A. Campbell, Campbell, H. E. A., David L. Wehlau +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Finite Group Theory Research #math.RT #msc:13A50
paper · pdf · doi:10.48550/arxiv.1308.3710
arxiv created 2013/08/16 · arxiv updated 2013/08/20
We study the ring of invariants for a finite dimensional representation V of the group C2 of order 2 in characteristic 2. Let σ denote a generator of C2 and \x1,y1 …, xm,ym\ a basis of V^*. Then σ(xi) = xi, and σ(yi) = yi + xi. To our knowledge, this ring (for any prime p) was first studied by David Richman in 1990. He gave a first main theorem for (V2, C2), that is, he proved that the ring of invariants when p=2 is generated by \xi, Ni = yi2 + xiyi, tr(A) | 2 ≤ |A| ≤ m\ where A ⊂ \0,1\m, yA = y1a1 y2a2 ⋯ ymam and tr(A) = yA + (y1+x1)a1(y2+x2)a2 ⋯ (ym+xm)am. In this paper, we prove the second main theorem for (V2, C2), that is, we show that all relations between these generators are generated by relations of type I: ∑I ⊂ A xI tr(A-I) and of type II: tr(A) tr(B) = ∑L < I xI-L NL tr(I-L+J+K) + NI ∑L < J xJ-L tr(L+K) for all m. We also derive relations of type III which are simpler and can be used in place of the relations of type II.