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 $V^*$. 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 $\{x_i, N_i = y_i^2 + x_iy_i, tr(A) | 2 \le |A| \le m\}$ 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.