Abstract Nonsense

Crushing one theorem at a time

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.

The Square Root Function and its Relation to Irreducible Characters

Let G be a finite group and define the map \sqrt{\text{ }}:G\to\mathbb{N}\cup\{0\} by \sqrt{g}=\#\left\{h\in G:h^2=g\right\}. We call this the square root function on G. Then, our previous characterization of real,complex, and quaternionic irreps can be phrased in terms of \sqrt{\text{ }}. Indeed, let \rho^{(\alpha)}:G\to\mathcal{U}\left(\mathscr{V}\right) be an irrep of the finite group G. Then,

\displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\chi^{(\alpha)}(g)=c_\alpha\quad\quad\mathbf{(1)}

Where c_\alpha is 1,0, and -1 if \rho^{(\alpha)} is real, complex, and quaternionic respectively. Note that the notation c^{(\alpha)} is appropriate (in the sense that it only depends on \alpha), since being a real, complex, or quaternionic irrep is invariant under equivalence.

To derive our main result we need to know very little about \sqrt{\text{ }} itself. In fact, we only need the following basic theorem:

Theorem: Let \sqrt{\text{ }}:G\to\mathbb{N}\cup\{0\} be defined as above. Then, \sqrt{\text{ }} is a class function.

 

Proof: Let g,h\in G be arbitrary. Define then f:\left\{k\in G:k^2=g\right\}\to\left\{k\in G:k^2=hgh^{-1}\right\} by k\mapsto hkh^{-1}. This mapping is well defined (in the sense that the image of the map really lies in the codomain) since if k^2=g then \left(hkh^{-1}\right)=hk^2h^{-1}=hgh^{-1}. Evidently then f is injective (being the restriction of the inner automorphism i_h) and thus it suffices to prove that f is surjective. To see this suppose that k is an element of the codomain, then k^2=hgh^{-1} and so h^{-1}k^2h=g but, of course h^{-1}k^2h=\left(h^{-1}kh\right)^2 and so h^{-1}kh is an element of the codomain. But, a quick check shows then that f\left(h^{-1}kh\right)=k. Since k\in\text{codom }f was arbitrary the conclusion follows. Thus, since f is a bijection we may conclude that

\sqrt{g}=\#\left\{k\in G:k^2=g\right\}=\#\left\{k\in G:k^2=hgh^{-1}\right\}=\sqrt{hgh^{-1}}

Since g,h\in G were arbitrary we may conclude that \sqrt{\text{ }}\in\text{Cl}(G) as required. \blacksquare

With this theorem in mind we are now well-equipped to prove the main result of this post. Namely:

Theorem: Let G be a finite group and h\in G. Then,

  

\displaystyle \sqrt{h}=\sum_{\alpha\in\widehat{G}}c_\alpha\chi^{(\alpha)}(h)

 

Proof: Citing once again the formula \mathbf{(1)} we may conjugate both sides to obtain

\displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\overline{\chi^{(\alpha)}(g)}=c_\alpha

Then, multiplying both sides by \chi^{(\alpha)}(h) gives

\displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\chi^{(\alpha)}(h)\overline{\chi^{(\alpha)}(g)}=c_\alpha \chi^{(\alpha)}(h)

Then, summing over \widehat{G} and performing minimal manipulations to the left-hand gives

\displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}\sum_{\alpha\in\widehat{G}}\chi^{(\alpha)}(h)\overline{\chi^{(\alpha)}(g)}=\sum_{\alpha\in\widehat{G}}c_\alpha\chi^{(\alpha)}(h)

But, using the second orthogonality relation and denoting the conjugacy class of h by \mathcal{C}_h we may rewrite this as

\displaystyle \frac{1}{|G|}\sum_{g\in G}\sqrt{g}c(g,h)\frac{|G|}{\#\left(\mathcal{C}_h\right)}=\sum_{\alpha\in\widehat{G}}c_\alpha\chi^{(\alpha)}(h)

Which, of course may be rewritten as

\displaystyle \frac{1}{\#\left(\mathcal{C}_h\right)}\sum_{g\in\mathcal{C}_h}\sqrt{g}=\sum_{\alpha\in\widehat{G}}c_\alpha \chi^{(\alpha)}(h)

But, by our previous theorem we know that \sqrt{\text{ }} is a class function and so the left-hand side reduces to

\displaystyle \frac{1}{\#\left(\mathcal{C}_h\right)}\#\left(\mathcal{C}_h\right)\sqrt{h}=\sqrt{h}

The conclusion follows. \blacksquare

References:

1. Isaacs, I. Martin. Character Theory of Finite Groups. New York: Academic, 1976. Print.

2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

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March 24, 2011 - Posted by | Algebra, Group Theory, Representation Theory | , , , , , , , , ,

2 Comments »

  1. […] have seen, using our last post, that there is an intimate relation between the square root function and the notion of […]

    Pingback by Representation Theory: The Connection Between the Square Root Function and the Number of Self-Conjugate Irreps « Abstract Nonsense | March 25, 2011 | Reply

  2. […] is the square root function for . Thus, if one knew entirely the nature of the square root function on then one would know the […]

    Pingback by Representation Theory: The Number of Self-Conjugate Irreps On a Finite Group of Odd Order « Abstract Nonsense | March 27, 2011 | Reply


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