Abstract Nonsense

Crushing one theorem at a time

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.

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

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.

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

Character Table of S_3 Without Finding Irreducible Characters


Point of post: In this post we construct the character table of S_3 without having to actually find the irreducible characters.

Motivation

As was stated in our last post this post shall serve to show how nice the theory we’ve devoloped  can make the construction of character tables. In particular, it is often very unapparent percisely how to construct all the irreducible characters. It is just an odd coincidence that for S_3 the irreps are so obvious. So, we shall show that in this post except for the trivial irrep which requires no thought to construct we don’t even need to construct either of the other two characters.

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March 22, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , | 2 Comments

Burnside’s Theorem


Point of post: In this post we put together a lot of our rep theory to prove one of the fundamental (pure) group theoretic results amenable to the subject.

Motivation

In this post we finally use representation theory to prove something in pure group theory that is near impossible to do without representation theory. We have seen on our thread about solvable groups that every p-group is solvable. In this thread we prove Burnside’s Theorem an amazing generalization which says that every group of order p^aq^b where p and q are primes. As a corollary we will be able to conclude that every non-abelian simple group is divisible by three distinct primes which, of course, will eliminate a respectable amount of group orders for analyzing simplicity. This is really one of the most beautiful applications of representation theory.

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March 11, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , | 1 Comment

A ‘Lemma’ (pt. II)


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

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March 10, 2011 Posted by | Algebra, Representation Theory | , , , , , | 1 Comment

A ‘Lemma’ (Pt. I)


Point of post: In this post we will prove a lemma which shall prove to be absolutely necessary in proving Burnside’s lemma.

 

Motivation

The basic idea of this post is to use a variant of the left \ast-representation of G generated by an irrep \rho to create a mapping \omega:\text{irr}(G)\to\mathbb{C}. We shall then use this to show that the number \displaystyle \frac{\chi(g)}{\chi(e)}\#\left(\mathcal{C}_g\right) is algebraic. This will prove to be surprising important in our proof of Burnside’s theorem.

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March 10, 2011 Posted by | Algebra, Representation Theory | , , , , , | 1 Comment

The Center of a Character (Pt. I)


Point of post: In this post we discuss the notion the ‘center of a character’ and its relations to other concepts we’ve previously learned.

Motivation

In previous posts we’ve seen how we can take concepts from plain group theory and attempt to make an analogy for characters. In this post we extend this further by defining the ‘center’ of a character, which, as we shall see are just those g\in G for which |\chi(g)|=\chi(e). In other words, its just those g\in G for which the modulus of \chi(g) is maximized. This shall prove useful in the future, in particular it shall help us devolop enough machinery to prove Burnside’s Theorem–the classic use of representation theory.

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March 8, 2011 Posted by | Algebra, Representation Theory | , , , , | 4 Comments

Relation Between the Kernels of Characters and Normal Subgroups


Point of post: In this post we derive some fundamental relationships between the kernels of the characters of a finite group G and the set of all normal subgroups of G. In particular, we prove that every normal subgroup of G has the realization as the kernel of some character, and thus by previous theorem that every normal subgroup of G has a realization as the intersection of certain N^{(\alpha)}‘s.

Motivation

We saw in our last post that there is a particularly fruitful way to produce normal subgroups ofa finite group G. Namely, one can look at the kernels of the characters of G, which as was shown amounts (in the sense that all the necessary information is contained in) to finding the kernels of the irreducible characters of G. What we shall now show is that in fact, finding the kernels of the irreducible characters of a group G enables us to list off precisely what normal subgroups G has. In particular, we will see that every normal subgroup of G has a realization as the kernel of a certain character. We shall do this by constructing a certain character on the quotient group of G for which, when thought of a character of G, will have kernel equal to the normal subgroup.

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March 7, 2011 Posted by | Algebra, Group Theory, Representation Theory | , , , , , | 4 Comments

Second Orthogonality Relation For Irreducible Characters


Point of post: In this post we discuss the second orthogonality relation for the irreducible characters. In particular, we prove that \displaystyle \sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(g)\overline{\chi^{(\alpha)}(h)}=c(g,h)\#\left(\bold{C}_G(g)\right) where c(g,h) takes the value one if g and h are conjugate and zero otherwise and \bold{C}_G(g) is the centralizer of g in G.

Motivation

In the past we’ve seen that if we ‘fix \alpha,\beta\in\widehat{G} and let g ‘vary’ over G (in the form of the sum) that there is an interesting orthogonality relation. Namely, this is just the orthogonality relation \displaystyle \frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}(g)\overline{\chi^{(\beta)}(g)}=\delta_{\alpha,\beta}. In this post we explore what happens if the ‘fixing’ and ‘varying’ are reversed. In other words, we compute \displaystyle \sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(g)\overline{\chi^{(\alpha)}(h)}. This shall serve as an interesting tool inall that comes. We call this the second orthogonality relation for the irreducible characters.

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March 7, 2011 Posted by | Algebra, Representation Theory | , , , , | 5 Comments

Using Orthogonality Relations to Compute Convolutions of Characters and Matrix Entry Functions


Point of post: In this post we will use our past results about the pairwise inner product of characters and matrix entry functions to compute their convolutions.

Motivation

In past posts we obainted certain relations between the pairwise inner product (as elements of the group algebra) of matrix entry functions and irreudcible characters. We shall use these relations to compute the convolution of matrix entry functions with eachother and likewise for characters.

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March 2, 2011 Posted by | Algebra, Representation Theory | , , , | 4 Comments