with highest weight has weights , it is easy to see that the associated character satisfies This expression is a finite geometric series that can be simplified to This last expression is just the statement of the Weyl character formula for the SU(2) case. Actually, following Weyl's original analysis of the representation theory of compact groups, one can classify the representations entirely from the group perspective, without using Lie algebra representations at all. In this approach, the Weyl character formula plays an essential part in the classification, along with the Peter–Weyl theorem. The SU(2) case of this story is described here. Relation to the representations of SO(3) Note that either all of the weights of the representation are even (if is even) or all of the weights are odd (if is odd). In physical terms, this distinction is important: The representations with even weights correspond to ordinary representations of the rotation group SO(3). By contrast, the representations with odd weights correspond to double-valued (spinorial) representation of SO(3), also known as projective representations. In the physics conventions, being even corresponds to being an integer while being odd corresponds to being a half-integer. These two cases are described as integer spin and half-integer spin, respectively. The representations with odd, positive values of are faithful representations of SU(2), while the representations of SU(2) with non-negative, even are not faithful. Another approach See under the example for Borel–Weil–Bott theorem. Most important irreducible representations and their applications Representations of SU(2) describe non-relativistic spin, due to being a double covering of the rotation group of Euclidean 3-space. Relativistic spin is described by the representation theory of SL2(C), a supergroup of SU(2), which in a similar way covers SO+(1;3), the relativistic version of the rotation group. SU(2) symmetry also supports concepts of isobaric spin and weak isospin, collectively known as isospin. The representation with (i.e., in the physics convention) is the 2 representation, the fundamental representation of SU(2). When an element of SU(2) is written as a complex matrix, it is simply a multiplication of column 2-vectors. It is known in physics as the spin-½ and, historically, as the multiplication of quaternions (more precisely, multiplication by a unit quaternion). This representation can also be viewed as a double-valued projective representation of the rotation group SO(3). The representation with (i.e., ) is the 3 representation, the adjoint representation. It describes 3-d rotations, the standard representation of SO(3), so