Abstract Nonsense

Crushing one theorem at a time

Induced Class Functions and the Space of Integral Class Functions (Pt. II)


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

\text{ }

Continue reading

Advertisements

April 27, 2011 Posted by | Algebra, Representation Theory | , , , , , | Comments Off on Induced Class Functions and the Space of Integral Class Functions (Pt. II)

The Connection Between the Square Root Function and the Number of Self-Conjugate Irreps (Cont.)


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

Continue reading

March 25, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , , , , , , | 1 Comment

The Connection Between the Square Root Function and the Number of Self-Conjugate Irreps


Point of post: In this post we find a formula for the number of \alpha\in\widehat{G} such that \rho^{(\alpha)} is self-conjugate for any \alpha\in\widehat{G}. As a by-product we must define the notion of an ambiguous conjugacy class in a group, and derive a relationship between the number of ambiguous subsets and the square root function.

Motivation

We have seen, using our last post, that there is an intimate relation between the square root function \sqrt{\text{ }}:G\to\mathbb{N}\cup\{0\} and the notion of self-conjugate representation. Indeed, we have seen that an irrep \rho^{(\alpha)}:G\to\mathcal{U}\left(\mathscr{V}\right) is self-conjugate if and only if \displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\chi^{(\alpha)}(g)=\pm 1. In this post we show that the relation deepens by showing that one can explicitly calculate the number of \alpha\in\widehat{G} such that \rho^{(\alpha)} is a self-conjugate irrep for any \alpha\in\widehat{G}  in terms of \sqrt{\text{ }}. Incidentally, we shall show there is another way to count the number of self-conjugate \alpha of a group by matching them up with the number of ‘ambivalent’ conjugacy classes in a group. Ambivalent conjugacy classes got their namesake via their apparent ambivalent nature as to whether they want to be subgroups in the sense that they are closed under inversion but not necessarily under multiplication. In connecting ambivalent subsets and the number of self-conjugate irreps we will inevitably show a relationship between the number of ambivalent subsets of the group and the square root function.

Continue reading

March 25, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , , , , , , | 2 Comments

The Square Root Function and its Relation to Irreducible Characters


Point of post: In this post we describe what can best be verbally described as “the number of square roots” function for a group and a way which it relates to the irreducible characters of the group.

Motivation

Recall that in our last post that we found an interesting property involving the characters: namely, we characterized real, complex, and quaternionic irreps in terms of their character. In this characterization a sum come up whose summand had the form \chi\left(g^2\right). That said, since the major theorems we have thus far developed involve summands of the form \chi(g) it would, of course, be preferable to change the summand in our characterization of real, complex, and quaternionic irreps into a summand involving \chi(g). The way we can do this is clear, namely for each h\in G we define the ‘square root’ of h, denoted \sqrt{h}, to be equal to \#\left\{g\in G:g^2=h\right\}. Then, with this it’s clear that our characterization can be rewritten as a sum with summand \sqrt{h}\chi(h). It turns out though that the interplay goes much farther than this, to the point where we can actually express \sqrt{h} entirely in terms of irreducible characters…and thus make it possible to compute \sqrt{h} from a groups character table.

Continue reading

March 24, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , , , , , , | 2 Comments

Irreducible Characters (Pt. II)


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

Continue reading

February 25, 2011 Posted by | Algebra, Representation Theory | , , , , , | 10 Comments

Irreducible Characters


Point of post: In this post we discuss what is arguably one of the most important tools in all of basic representation theory– the irreducible characters of a group.

Motivation

In our last few posts we’ve been developing the notion of a class function and the space of class functions under the hazy motivation that we will use them to ascertain that the cardinality of some set X is the dimension of \text{Cl}(G) (we, in our last post, showed that this was the number k of conjugacy classes in G). In this set we shall see that this set X for which we’d like to show has the quality \#(X)=\dim\text{Cl}(G) are the irreducible characters of the group G. These are certain class functions which will occupy a fair amount of our efforts in the coming posts since, in a very real sense, they make up a vast portion of the substance in basic representation theory. But, for now we shall restrict our attention to defining the irreducible characters and showing that they form an orthonormal basis for \text{Cl}(G). Of this we shall get the corollary that \#\left(\widehat{G}\right)=k.

Continue reading

February 25, 2011 Posted by | Algebra, Representation Theory | , , , , , | 10 Comments

Dimension of the Space of Class Functions


Point of post: In this post we prove the simple result that the dimension of the space of class functions is equal to the number of conjugacy classes in G.

Motivation

In our last post we hinted that the dimensionality of the space \text{Cl}(G) of class functions of the finite group G shall be used to derive a very interestint result. As a step toward this we prove in this post that the dimension of \text{Cl}(G) thought of as a subspace of the group algebra \mathcal{A}(G) is the number of conjugacy classes of G .

Continue reading

February 24, 2011 Posted by | Algebra, Representation Theory | , , , , | 2 Comments

Class Functions


Point of post: In this post we derive results about the set of class functions on a finite group G, in particular finding its dimension as a subspace of the group algebra and characterizing it as the center of the group algebra.

Motivation

In our last series of posts we saw an interesting technique. We saw the interesting idea that if we want to prove the cardinality of a set X is equal to \kappa it suffices to construct a vector space \mathscr{V} of dimension \kappa such that X is a basis for \mathscr{V}. In particular, we saw that \displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=|G| by considering the group algebra \mathcal{A}(G) of dimension |G| and then showing that \left\{D^{(\alpha)}_{i,j}:\alpha\in\widehat{G}\text{ and }i,j\in[d_\alpha]\right\} is a basis for \mathcal{A}(G) (where, as in the last post, the D^{(\alpha)}_{i,j} are the matrix entry functions). We now wish to get our milage out of this technique by applying it again in a different context (different set and different cardinality). We don’t want to ruin the surprise of what precisely this will be, but we shall construct the vector space with the ‘proper dimension’ in this post. In particular, we will consider and study the set of class functions on a finite group G. Intuitively, these are functions which satisfy (of course we mean this loosely since we haven’t define the domain, range, etc.) the common ‘trace identity’ \text{tr}\left(ABA^{-1}\right)=\text{tr}(B).

Continue reading

February 24, 2011 Posted by | Algebra, Representation Theory | , , , , | 14 Comments