Abstract Nonsense

Crushing one theorem at a time

The Number of Self-Conjugate Irreps On a Finite Group of Odd Order


Point of post: In this post we use our recent works on the number of self-conjugate irrep classes of a finite group to show that for finite groups of odd order every non-trivial irrep is complex.

Motivation

We’ve done much recent work on finding different characterizations of the number of self-conjugate irrep classes of a finite group. In particular, we’ve show that if G is a finite group and \mathfrak{s} denotes the number of self-conjugate irreps classes of G then

 

\displaystyle \mathfrak{s}=\frac{1}{|G|}\sum_{g\in G}\sqrt{g}\;^2

 

where \sqrt{\text{ }} is the square root function for G. Thus, if one knew entirely the nature of the square root function on G then one would know the nature of the number of self-conjugate irrep classes of G. Unfortunately, it is hard to say anything about the square root function on an entirely general group–in particular there is no way to calculate the square root function if G is a finite group of even order. That said, as we shall see the square root function has a particularly simple description on finite groups of odd order. Pursuant to this simple nature we shall prove that if G is a finite group of odd order then the only self-conjugate irrep class of G is the trivial irrep class \alpha_\text{triv}.

 

Continue reading

Advertisements

March 27, 2011 Posted by | Algebra, Representation Theory | , , , , , , , | 1 Comment

A Characterization of Real, Complex, and Quaternionic Irreps


Point of post: In this post we derive a result historically attributed to Frobenius and Schur which gives us a characterization to real, complex, and quaternionic irreps based on their admittant characters.

Motivation

In the past we’ve discussed how the set of all irreps are naturally carved up into three subclasses: real, complex, and quaternionic. This analogizes the difference between real, complex, and quaternionic numbers. It turns out that in general it is not, at first glance, clear how to determine from elementary methods whether or not an irrep was real, complex, or quaternionic. Indeed, in our one example of quaternionic irreps the agrument that the irrep in question was, in fact, quaternionic was involved and admittedly convoluted. That said, the theorem we develop in this post shall give us a simple way to determine whether an irrep is real, complex, and conjugate by a simple calculation involving the character of the irrep.

Continue reading

March 23, 2011 Posted by | Algebra, Representation Theory | , , , , , , | 2 Comments