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?
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$\begingroup$ The groups acts by conjugation on $A_iA_j^{-1}$, so one has the conjugacy invariants of these matrices and their products (such as the coefficients of the characteristic polynomial of $A_iA_j^{-1}$). I suspect these and the determinants of $A_i$ generate the ring of invariants, but I wouldn't know how to prove this. A paper of Procesi may be relevant: ams.org/journals/bull/1976-82-06/S0002-9904-1976-14196-1/… $\endgroup$– Ian AgolSep 16, 2012 at 23:41
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$\begingroup$ Might be Ed Formanek (mathoverflow.net/users/9347/ed-formanek) is right person to ask if matrices are not symmetric he gave wonderful answer here: mathoverflow.net/questions/85664/… refering to his CBMS notes "The Polynomial Identities and Invariants of matrices". $\endgroup$– Alexander ChervovSep 17, 2012 at 7:05
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3 Answers
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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.
answered Sep 17, 2012 at 7:33
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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.
answered Sep 17, 2012 at 6:23
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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.
answered Sep 17, 2012 at 14:18