From: Dmitrii Pasechnik
Newsgroups: sci.math.research
Subject: Re: covariants
Date: 14 Apr 1998 15:29:55 +0200
Allen Adler writes:
> Let $d_1,\dots,d_n$ be positive integers. For every nonnegative
> integer $m$, let $V_m$ denote the symmetric tensor representation
> of degree $m$ of $G=SL(2,C)$ and let $X$ be the tensor product
> of the $n$ representations $V_m$ with $m=d_1,\dots,d_n$.
>
> Let $A=A(d_1,\dots,d_n)$ denote the ring of invariants for $G$ acting
> on $X$.
>
> I would like to know of all cases in which the following information
> is known:
> (1) An explicit homogeneous system of parameters for $A$;
> (2) An explicit basis for $A$ over the subalgebra generated
> by the homogeneous system of parameters.
>
> I'm aware of examples in which generators are known for the ring
> of invariants and even where the basic syzygies are known. But
> I don't know of more detailed presentations of the type I have
> asked for above except in very trivial cases.
>
It seems that the common view is that
computing the system of parameters is as hard as computing the
generators themselves
I presume that all known up to now can be found in in the following
two references I append.
A slightly different question: suppose that a system of parameters for
the invariants of a maximal torus T