Abstract Nonsense

Crushing one theorem at a time

A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces


Point of post: In this post we discuss a way of taking a \mathbb{C}-representation and using it to create another representation which satisfies the real condition. Moreover, when the original \mathbb{C}-representation is an irrep which is quaternionic or complex then the resulting representation will have no non-trivial (\rho,J)-invariant subspaces.

Motivation

In our last post we have seen that \mathbb{C}-representations satisfying the real condition are important since they correspond to real representations in a natural way. Moreover, we have seen that \mathbb{C}-representations satisfying the real condition with realizer J with no non-trivial proper \left(\rho,J\right)-invariant subspaces are even more interesting since they correspond naturally to irreducible real representations. We shall now show a method which takes a general \mathbb{C}-representations and produces a \mathbb{C}-representation which satisfies the real condition. Moreover, we shall see that if the original irrep happens to be complex or quaternionic irrep then the corresponding representation won’t have any non-trivial proper (\rho,J)-invariant subspaces.

 

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April 2, 2011 Posted by | Algebra, Representation Theory | , , , , | 3 Comments

A Bijection Between A Subset of The Complex Reps of a Finite Group and the Real Reps (Pt. III)


Point of post: This is a continuation of this post.

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March 30, 2011 Posted by | Algebra, Representation Theory | , , , , | 2 Comments

A Bijection Between A Subset of the Complex Reps of a Finite Group and the Real Reps (Pt. II)


Point of post: This post is a continuation of this one.

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March 30, 2011 Posted by | Algebra, Representation Theory | , , , | 1 Comment