Saturday, 15 September 2012

rt.representation theory - irreducible subgroup of SL(n,R)

A representation rho over a field K is called absolutely irreducible if for any algebraic field extension L/K, the representation rhootimesKL obtained by extension of scalars is irreducible (over L). It is enough to check this for the algebraic closure. As damiano's examples in the comments show, this is a much stronger property than irreducibility. Serre's "Linear representations of finite groups" contains a criterion for a real representation of a finite group to be absolutely irreducible.



Here is a way in which non absolutely irreducible representations typically arise. Let L/K be a finite separable field extension of degree d>1 and sigma be an irreducible n-dimensional representation of G over L. By restriction of scalars, we obtain an nd-dimensional representation rho of G over K. (In the language of linear group actions, the representation space, which is a vector space over L, is viewed as a vector space over K). The representation rho is not absolutely irreducible because rhootimesKL is isomorphic to the direct sum of d>1 Galois conjugates of sigma. Yet rho is frequently irreducible. For example, under the restriction of scalars from mathbbC to mathbbR, the group U(1,mathbbC) becomes SO(2,mathbbR). Therefore, any one-dimensional complex unitary representation (i.e. a character) sigma of a group G gives rise to a two-dimensional real orthogonal representation rho whose complexification splits into a direct sum of two representations. This is the construction behind damiano's second and third examples.

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