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.

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