循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of FV]～G.We prove that FV]～G = Fl_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.

Modular Vector Invariants of Cyclic Groups $Z_{2}$

Let $G$ be the finite cyclic group $Z_{2}$ and $V$ be a vector space of dimension $2n$ with basis $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ over the field $F$ with characteristic 2. If $\sigma$ denotes a generator of $G$, we may assume that $\sigma(x_{i})=ay_{i}$, $\sigma(y_{i})=a^{-1}x_{i}$, where $a\in F^{*}$. In this paper, we describe the explicit generator of the ring of modular vector invariants of $FV]^{G}$. We prove that $$FV]^{G}=Fl_{i}=x_{i}+ay_{i}, q_{i}=x_{i}y_{i},1\leq i\leq n,M_{I}=X_{I}+a^{|I|}Y_{I}],$$ where $I\subseteq A_{n}=\{1,2,\ldots,n\}$, $2\leq |I|\leq n$.