Invariants of group action: SL_n acts simultaneously on m symmetric matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:27:12Z http://mathoverflow.net/feeds/question/107340 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107340/invariants-of-group-action-sl-n-acts-simultaneously-on-m-symmetric-matrices Invariants of group action: SL_n acts simultaneously on m symmetric matrices Döni 2012-09-16T19:20:28Z 2012-09-17T14:18:58Z <p>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? </p> http://mathoverflow.net/questions/107340/invariants-of-group-action-sl-n-acts-simultaneously-on-m-symmetric-matrices/107361#107361 Answer by Dima Pasechnik for Invariants of group action: SL_n acts simultaneously on m symmetric matrices Dima Pasechnik 2012-09-17T06:23:35Z 2012-09-17T06:23:35Z <p>For $n=2$ an answer is classically known, see Grace and Young, <a href="http://archive.org/stream/algebraofinvaria00graciala#page/161/mode/1up" rel="nofollow">p.161, 139A</a>. Probably there is a modern explanation of this result known, too.</p> http://mathoverflow.net/questions/107340/invariants-of-group-action-sl-n-acts-simultaneously-on-m-symmetric-matrices/107363#107363 Answer by Bruce Westbury for Invariants of group action: SL_n acts simultaneously on m symmetric matrices Bruce Westbury 2012-09-17T07:33:09Z 2012-09-17T10:28:35Z <p>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.</p> <p>Apply the theory in the following paper to the quiver with one vertex and $m$ edges.</p> <p>MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio</p> <p>Semisimple representations of quivers.</p> <p>Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.</p> <p>The special case you are considering was studied before this. The reference has already been given by Agol.</p> http://mathoverflow.net/questions/107340/invariants-of-group-action-sl-n-acts-simultaneously-on-m-symmetric-matrices/107384#107384 Answer by Abdelmalek Abdesselam for Invariants of group action: SL_n acts simultaneously on m symmetric matrices Abdelmalek Abdesselam 2012-09-17T14:18:58Z 2012-09-17T14:18:58Z <p>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 <a href="http://arxiv.org/abs/0805.4135" rel="nofollow">La théorie des invariants des formes quadratiques ternaires revisitée''</a> by Bruno Blind.</p>