## 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 such that is self-conjugate for any . 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 and the notion of self-conjugate representation. Indeed, we have seen that an irrep is self-conjugate if and only if . In this post we show that the relation deepens by showing that one can *explicitly *calculate the number of such that is a self-conjugate irrep for any in terms of . Incidentally, we shall show there is another way to count the number of self-conjugate 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.

*The Number of Self-Conjugate Irreps of a Finite Group*

To save notation, we recall that being real, complex, or quaterinonic as an irrep is invariant under equivalence (althought I actually neglected to prove this when I first discussed real, complex, and quaternionic irreps since equivalent irreps admit the same irreducible character this follows immediately from our recent characterization). Thus, for any one has that the adjectives real, complex, quaternionic, and self-conjugate apply simultaneously to every element of .. Thus, it makes sense to define to be real, complex, quaternionic, or self-conjugate if any one of its elements is real, complex,quaternionic, or self-conjugate respectively. We define then, in accordance with previous definitions, the function with denoted to be and if is real, complex, or quaternionic respectively. With this in mind we proceed to derive our first result. Namely:

**Theorem: ***Let be a finite group and the number of self-conjugate . Then,*

* *

* *

**Proof: **Recall from a previous result that

Consequently,

** **

** **

From where the conclusion follows.

**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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