4-2017

Honors Thesis

#### Degree Name

Bachelors of Science (BS)

Mathematics

C. Ryan Vinroot

Gexin Yu

Marc Sher

#### Abstract

A group $G$ is called \emph{real} if every element is conjugate to its inverse, and $G$ is \emph{strongly real} if each of the conjugating elements may be chosen to be an involution, an element in $G$ which squares to the identity. Real groups are called as such because every irreducible character of a real group is real valued. A group $G$ is called \emph{totally orthogonal} if every irreducible complex representation is realizable over the field of real numbers. Total orthogonality is sufficient, but not necessary for reality.

Reality of representations is quantified in the Frobenius-Schur indicator. For an irreducible character $\chi$ of $G$, the Frobenius-Schur indicator is given by

$\varepsilon(\chi) = \frac{1}{|G|}\sum_{g \in G} \chi(g^2).$

Frobenius and Schur showed that

$\varepsilon(\chi) = \begin{cases} 1 & \text{if } \chi \text{ is the character of a real representation} \\ -1 & \text{if }\chi \text{ is real-valued but is not the character of a real representation} \\ 0 & \text{otherwise} \end{cases}.$

They also showed that

$\sum_{\chi \in \Irr(G)} \varepsilon(\chi)\chi(1) = \big | \{g \in G : g^2 =1 \} \big |,$

where $\Irr(G)$ is the collection of irreducible characters of $G$. Hence a group is totally orthogonal if and only if its character degree sum is equal to the number of involutions in the group. It is conjectured that a finite simple group is strongly real if and only if it is totally orthogonal.

In this work, we verify this conjecture for all strongly real simple groups except $\PSp(2n,q)$ and $\POmega^\pm(4n,q)$ when $q$ is even. The methods in this paper reduce this conjecture to showing that $\Or^\pm(4n,q)$ and $\Sp(2n,q)$ are totally orthogonal.

Motivated by the Frobenius-Schur count of involutions and Conjecture 7.1 of \cite{vinroot}, we prove an upper bound on the number of involutions in $\Or^\pm(2n,q)$ and $\Sp(2n,q)$ by $q^{(d-r)/2}(q+1)^r$ where $r$ is the rank of the group and $d$ is its dimension. If indeed, $\Or^\pm(4n,q)$ and $\Sp(2n,q)$ are totally orthogonal, this verifies Conjecture 7.1 of \cite{vinroot} for these groups.

Finally, we obtain generating functions for the number of involutions in subgroups of the orthogonal groups. We apply these generating functions to compute the limiting behavior of the number of involutions in these groups as the authors did in \cite{fulman} for some other classical groups.

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