A useful semistability criterion (Q2718951)

In this short note, the author gives a useful criterion for checking the GIT (= geometric invariant theory) semi-stability of points. Let \(G\) be a reductive group acting linearly on a linear space \(W\), thus acting on \(\mathbb P(W)\). The GIT theory tells us that there is an open \(G\)-invariant subset \(\mathbb P(W)^{ss}\) so that a GIT quotient \(\mathbb P(W)^{ss}//G\) exists and is projective. The stability criterion is a simple rule that allows one to identify which points are in \(\mathbb P(W)^{ss}\). NEWLINENEWLINENEWLINEIn this article, the author considers the case when \(W=\bigoplus^n W_i\) and the linear action of \(G\) on \(W\) is induced by its action on individual pieces \(W_i\). The criterion stated in this note reduces the stability of a point \([w_1,\cdots,w_n]\in \mathbb P(W)\) to that of \([w_i]\in \mathbb P(W_i)\). This will be useful in studying some moduli problems.