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EDIT: Wrong definition of $\beta\left(G,H\right)$ fixed. One of the results is open (i. e., I cannot prove it).


In "Finite Groups and invariant theory" (a paper in Malliavin's LNM #1478 which can be found on Springerlink and in other sources), Barbara Schmid defines a positive integer $\beta\left(G\right)$ for any finite group $G$ and any field $K$ of characteristic $0$ as the smallest integer $k$ such that for any representation $V$ of $G$ over $K$, the invariant ring $K\left[V\right]^G$ is generated by the invariants of degree $\leq k$.

Here is my very first question: Is there a simple reason why $\beta\left(G\right)$ is stable under field extension of $K$ ? I can prove that it is stable under field extension of $K$ for any single representation of $G$ (because being invariant under $G$ means satisfying a family of linear equations, and being generated by invariants of smaller degree means satisfying another family of linear equations), but the problem is that some representations only pop up when the field is sufficiently extended. The problem can be avoided by referring to Larry Smith's note which shows that the regular representation is the worst case (and it is, of course, defined over every field), but this altogether leads to a not particularly direct proof. (EDIT: There is a simpler proof: whenever $V$ is a subrepresentation of another representation $W$, the $\beta\left(G\right)$ number evaluated at $V$ is smaller or equal to that evaluated at $W$. Now use some basic Galois theory.)

Schmid proves some nice results on $\beta$:

  • Noether's bound says that $\beta\left(G\right)\leq \left|G\right|$. (Larry Smith showed that this actually holds in the $\mathrm{char} K\nmid \left|G\right|$ case, but for me the $\mathrm{char} K=0$ case is enough at the moment.)

  • If $A$ is a commutative filtered $K$-algebra with filtration $A_0\subseteq A_1\subseteq A_2\subseteq ...$ such that $A_0=K$ and such that $A$ is generated by $A_1$ over $K$, and if $G$ acts on $A$ by $K$-algebra automorphisms respecting the filtration, then the ring of invariants $A^G$ is generated (as a $K$-algebra) by the invariants in $A_{\beta\left(G\right)}$. (This is trivial (by the very definition of $\beta\left(G\right)$) whenever $A$ is the coordinate ring of a representation, but here we don't require this.)

  • If $H$ is a subgroup of $G$, then $\beta\left(G\right)\leq \left[G:H\right]\beta\left(H\right)$.

  • If $N$ is a normal subgroup of $G$, then $\beta\left(G\right)\leq \beta\left(G\diagup N\right)\beta\left(N\right)$.

Now I think this $\beta$ notion has a relative version. My question is: Is it any good? Has it been studied?

For a group $G$ and a subgroup $H\subseteq G$, we define a number $\beta\left(G,H\right)$ as the smallest integer $k$ such that for any representation $V$ of $G$ over $K$, and for any system $\left(u_1,u_2,...\right)$ of $K$-algebra generators of the invariant ring $K\left[V\right]^H$, the invariant ring $K\left[V\right]^G$ is generated by invariants which can be written as polynomials of degree $\leq k$ in the variables $gu_i$ for $g\in G$ and $i\in\left\lbrace 1,2,3,...\right\rbrace$.

Similarly to Noether's bound, we can show that $\beta\left(G,H\right)\leq \left[G:H\right]$.

I can prove the following facts (in a rather ugly way):

  • If $A$ is a commutative filtered $K$-algebra with filtration $A_0\subseteq A_1\subseteq A_2\subseteq ...$ such that $A_0=K$ and such that $A$ is generated by $A_1$ over $K$, and if $G$ acts on $A$ by $K$-algebra automorphisms respecting the filtration, and if the ring of invariants $A^H$ is generated (as a $K$-algebra) by the invariants in $A_1$, then the ring of invariants $A^G$ is generated (as a $K$-algebra) by the invariants in $A_{\beta\left(G,H\right)}$.

  • If $H^{\prime}\subseteq H\subseteq G$ is a subgroup tower, then $\beta\left(G,H^{\prime}\right)\leq \beta\left(G,H\right)\beta\left(H,H^{\prime}\right)$.

  • If $N$ is a normal subgroup of $G$, then $\beta\left(G,N\right)=\beta\left(G\diagup N\right)$.

I am wondering whether any subgroup tower $H^{\prime}\subseteq H\subseteq G$ must satisfy $\beta\left(G,H\right)\leq \beta\left(G,H^{\prime}\right)$. And I am wondering whether the above results are any good, except of giving a rather involved generalization of the standard $\beta$. Is there a reasonable way to compute $\beta\left(G,H\right)$ ? (It sounds way harder than computing $\beta\left(G\right)$. Can this relative $\beta$ be of any use in proving results about the absolute $\beta$ ?

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