For example, the following real representation of the cyclic group is reducible when considered over Therefore, by classifying all the irreducible representations that are real over we still haven't classified all the irreducible real representations. But we achieve the following: Let be a real vector space. Let act irreducibly on and let If is not irreducible, there are exactly two irreducible factors which are complex conjugate representations of Definition. A quaternionic representation is a (complex) representation which possesses a –invariant anti-linear homomorphism satisfying Thus, a skew-symmetric, nondegenerate –invariant bilinear form defines a quaternionic structure on Theorem. An irreducible representation is one and only one of the following: (i) complex: is not real-valued and there exists no –invariant nondegenerate bilinear form on (ii) real: a real representation; has a –invariant nondegenerate symmetric bilinear form. (iii) quaternionic: is real, but is not real; has a –invariant skew-symmetric nondegenerate bilinear form. Representations of particular groups Symmetric groups Representation of the symmetric groups have been intensely studied. Conjugacy classes in (and therefore, by the above, irreducible representations) correspond to partitions of n. For example, has three irreducible representations, corresponding to the partitions 3; 2+1; 1+1+1 of 3. For such a partition, a Young tableau is a graphical device depicting a partition. The irreducible representation corresponding to such a partition (or Young tableau) is called a Specht module. Representations of different symmetric groups are related: any representation of yields a representation of by induction, and vice versa by restriction. The direct sum of all these representation rings inherits from these constructions the structure of a Hopf algebra which, it turns out, is closely related to symmetric functions. Finite groups of Lie type To a certain extent, the representations of the , as n varies, have a similar flavor as for the ; the above-mentioned induction process gets replaced by so-called parabolic induction. However, unlike for , where all representations can be obtained by induction of trivial representations, this is not true for . Instead, new building blocks, known as cuspidal representations, are needed. Representations of and more generally, representations of finite groups of Lie type have been thoroughly studied. describes the representations of . A geometric description of irreducible representations of such groups, including the above-mentioned cuspidal representations, is obtained by Deligne-Lusztig theory, which constructs such representation in the l-adic cohomology of Deligne-Lusztig varieties. The similarity of the representation theory of and goes beyond finite