Fundamental representation → how symmetry acts on matter Adjoint representation → how symmetry acts on itself, it is one of the most profound structural facts in physics.

We know the SU(2) The symmetry rotates the components of the matter field, components include Electron spin, weak isospin doublets or any SU(2) doublet.

Now in adjoint representation, the components in the space if not spinor but vector.

he “vector” is:$$X = X^a T_a$$

So the coefficients $X^a$Xa form a 3-vector.

Adjoint rep is that the Symmetry rotates the symmetry directions themselves.