The Second Main Theorem Vector for the modular regular representation of $C_2$

Abstract

We study the ring of invariants for a finite dimensional representation $V$ of the group $C_2$ of order 2 in characteristic $2$. Let $\sigma$ denote a generator of $C_2$ and $\{x_1,y_1\dots ,x_m,y_m\}$ a basis of $V^{\ast }$. Then $\sigma (x_i)=x_i$, and $\sigma (y_i)=y_i+x_i$. To our knowledge, this ring (for any prime $p$) was first studied by David Richman in 1990. He gave a first main theorem for $(V_2,C_2)$, that is, he proved that the ring of invariants when $p=2$ is generated by $\{x_i,N_i=y_i^2+x_i{y}_i,tr(A)|2\le |A|\le m\}$ where $A\subset \{0,1{\}}^m$, $y^A=y_1^{a_1}{y}_2^{a_2}\cdots y_m^{a_m}$ and $tr(A)=y^A+(y_1+x_1{)}^{a_1}(y_2+x_2{)}^{a_2}\cdots (y_m+x_m{)}^{a_m}.$ In this paper, we prove the second main theorem for $(V_2,C_2)$, that is, we show that all relations between these generators are generated by relations of type I: ${\sum }_{I\subset A}{x}^{I}tr(A-I)$ and of type II: $tr(A)tr(B)={\sum }_{L<I}{x}^{I-L}{N}^{L}tr(I-L+J+K)+N^I{\sum }_{L<J}{x}^{J-L}tr(L+K)$ 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.