Degree bounds for modular covariants

Modular Covariants of Cyclic Groups of Order p

Let $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module …

Modules of covariants in modular invariant theory

Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\subset k[V]^H$ is studied using modules of covariants. An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the reflections in $G$. A classical theorem due to Serre says that if $k[V]$ is a free $k[V]^G$-module then $G=W$. We generalize this result as follows. If $k[V]^H$ is a free $k[V]^G$-module then $G$ is generated by $H$ and $W$, and the invariant ring $k[V]^{H\cap W}$ is free over $…

Degree of reductivity of a modular representation

For a finite dimensional representation $V$ of a group $G$ over a field $F$, the degree of reductivity $δ(G,V)$ is the smallest degree $d$ such that every nonzero fixed point $v\in V^{G}\setminus\{0\}$ can be separated from zero by a homogeneous invariant of degree at most $d$. We compute $δ(G,V)$ explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian $p$-groups.

On complete system of covariants for the binary form of degree 7

A minimal system of 147 homogeneous generating elements of the algebra of covariants for the binary form of degree 7 is calculated.

The degree of the algebra of covariants of a binary form

We calculate the degree of the algebra of covariants $\mathcal{C}_d$ for binary $d$-form. Also, for the degree we obtain its integral representation and asymptotic behavior.

模不变量的通用分离

Generic separation for modular invariants

针对素数阶循环群的模表示，提出低次多项式不变量与次数$p$不变量，实现通用轨道分离并生成有理不变量域，并推广至可分解表示。