The second main theorem vector for the modular regular representation of C2C2

We study the ring of invariants for a finite dimensional representation V   of the group C2C2 of order 2 in characteristic 2. Let σ   denote a generator of C2C2 and {x1,y1,…,xm,ym}{x1,y1,…,xm,ym} a basis of V⁎V⁎. Then σ(xi)=xiσ(xi)=xi, and σ(yi)=yi+xiσ(yi)=yi+xi.To our knowledge, this ring (for any prime p) was first studied by David Richman [12] in 1990. He gave a first   main theorem for (V2,C2)(V2,C2), that is, he proved that the ring of invariants when p=2p=2 is generated by{xi,Ni=yi2+xiyi,tr(A)|2⩽|A|⩽m}, where A⊂{0,1}mA⊂{0,1}m, yA=y1a1y2a2⋯ymam andtr(A)=yA+(y1+x1)a1(y2+x2)a2⋯(ym+xm)am.tr(A)=yA+(y1+x1)a1(y2+x2)a2⋯(ym+xm)am. In this paper, we prove the second   main theorem for (V2,C2)(V2,C2), that is, we show that all relations between these generators are generated by relations of type I∑I⊂AxItr(A−I)=0, and of type IItr(A)tr(B)=∑L