SL(2,C) is the special linear group of 2x2 complex matrices with determinant 1. It is important in representation theory because it is the simplest example of a non-abelian Lie group, and its representations can be used to understand representations of more complicated Lie groups.
An irreducible representation of a group is a linear representation that cannot be decomposed into smaller, invariant subspaces. In other words, it is a representation that does not have any non-trivial subrepresentations.
The irreducible representations of SL(2,C) can be determined using the theory of highest weight representations. This involves finding a set of generators for the Lie algebra of SL(2,C) and then constructing representations from these generators.
There is a close relationship between SL(2,C) and SO(3), as SL(2,C) is the universal cover of SO(3). This means that every irreducible representation of SO(3) can be lifted to an irreducible representation of SL(2,C), but not vice versa.
Irreducible representations of SL(2,C) are used in physics to study the symmetries of physical systems, particularly in quantum mechanics and particle physics. For example, the spin of a particle can be described using an irreducible representation of SL(2,C).