Invariants of group action: SL_n acts simultaneously on m symmetric matrices - MathOverflow

Invariants of group action: SL_n acts simultaneously on m symmetric matrices

2012-09-16T19:20:28Z

Let $\rm{SL}_n$ be the special linear group and let $\rm{Sym}_n$ be the set of all symmetric matrices of size n. $\rm{SL}_n$ acts on $(\rm{Sym}_n)^m$ by $g(A_1, \ldots , A_m)=(gA_1 g^{\rm T}, \ldots , g A_m g^{\rm T})$. Clearly, in the case of $m=1$ the ring of invariants is generated by $\det(A)$. But what are the invariants of this group action in general? Is there an easy description of the ring of invariants, e.g. by giving the generators?

Answer by Dima Pasechnik for Invariants of group action: SL_n acts simultaneously on m symmetric matrices

2012-09-17T06:23:35Z

For $n=2$ an answer is classically known, see Grace and Young, p.161, 139A. Probably there is a modern explanation of this result known, too.

Answer by Bruce Westbury for Invariants of group action: SL_n acts simultaneously on m symmetric matrices

2012-09-17T07:33:09Z

If you are willing to replace $SL$ by $GL$ (so without the determinant) then this is a special case of a much more general result.

Apply the theory in the following paper to the quiver with one vertex and $m$ edges.

MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio

Semisimple representations of quivers.

Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.

The special case you are considering was studied before this. The reference has already been given by Agol.

Answer by Abdelmalek Abdesselam for Invariants of group action: SL_n acts simultaneously on m symmetric matrices

2012-09-17T14:18:58Z

This is part of classical invariant theory: the study of joint invariants of several quadratic forms. As far as I know these rings of invariants are only known in a few special cases. There has been work by Turnbull and Todd for the $n=3$ case. A recent paper on the subject which contains pointers to the classical literature is ``La théorie des invariants des formes quadratiques ternaires revisitée'' by Bruno Blind.