§ Example for invariant theory

• Consider $p(z, w) = p_1 z^2 + p_2 zw + p_3 w^2$ --- binary forms of degree two.
• The group $SL(2, Z)$ acts on these by substituting $(z, w) \mapsto PSL(2, Z) (z, w)$.
• We can write the effect on the coefficents explicitly: $(p_1', p_2', p_3') = M (p_1, p_2, p_3)$.
• So we have a representation of $SL(2, Z)$.
• An example

• IAS lecture