= Hilbert-Mumford criterion for projective stability
{c}
{title2=$\mu([v],\lambda)=-\min\{w:v_w\ne0\}$}
For a complex <reductive algebraic group> acting linearly on $V$, decompose a nonzero vector into weights for each <algebraic one-parameter subgroup>. With the displayed sign convention, semistability is equivalent to $\mu\ge0$ for every such <subgroup>, and stability is equivalent to strict positivity for every nontrivial <subgroup>. One must test conjugates of the <subgroups> as well as those diagonal in a single fixed coordinate system.
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