# Abstract Nonsense

## 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.