From: Dmitrii Pasechnik <dima@duti515a.twi.tudelft.nl>
Newsgroups: sci.math.research
Subject: Re: covariants
Date: 14 Apr 1998 15:29:55 +0200

Allen Adler <adler@hera.wku.edu> 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<G is known. Could one use it
to compute a system of parameters for the G-invariants ?

Dmitrii.
d.pasechnik at twi dot tudelft dot nl

-----------------------------
@incollection {MR88e:15011,
   AUTHOR = {Dixmier, J.},
    TITLE = {Quelques r\'esultats et conjectures concernant les s\'eries de
             {P}oincar\'e des invariants des formes binaires},
BOOKTITLE = {S\'eminaire d'alg\`ebre Paul Dubreil et Marie-Paule Malliavin,
             36\`eme ann\'ee (Paris, 1983--1984)},
    PAGES = {127--160},
   SERIES = {Lecture Notes in Math.},
   VOLUME = {1146},
PUBLISHER = {Springer},
  ADDRESS = {Berlin},
     YEAR = {1985},
}
@article {MR92d:16032,
   AUTHOR = {Littelmann, P. and Procesi, C.},
    TITLE = {On the {P}oincar\'e series of the invariants of binary forms},
  JOURNAL = {J. Algebra},
 FJOURNAL = {Journal of Algebra},
   VOLUME = {133},
     YEAR = {1990},
   NUMBER = {2},
    PAGES = {490--499},
     ISSN = {0021-8693},
    CODEN = {JALGA4},
}