What is representation theory? A group G is an abstract symmetry such as rotations, boots, gauge transformations, phase rotations and diffeomorphisms, etc. but physics cannot act with abstract symbols, it needs objects that transform, so we need to use representation.

A representation is a rule that assigns to each group element $g$g a linear transformation acting on some vector space.

Concisely, the smallest eigenvalues of $J_z$Jz​ are 1/2. A weyl spinor (1/2, 0) means

It is spin-½ under the left SU(2); It is scalar under the right SU(2)