Application of Character Estimates to the Number of $T_2$-Systems of the Alternating Group

TL;DR

This paper employs character theory and estimates to derive a lower bound on the number of $T_2$-systems in the alternating group $A_n$, providing insights into the structure of the associated product replacement graph.

Contribution

It introduces a novel application of character estimates to quantify $T_2$-systems in $A_n$, linking group theory with graph connectivity analysis.

Findings

Established a lower bound for $T_2$-systems in $A_n$.

Derived a lower bound for connected components of the product replacement graph.

Demonstrated the effectiveness of character estimates in combinatorial group theory.

Abstract

We use character theory and character estimates to show that the number of $T_2$-systems of $A_n$ is at least \begin{equation*} \frac{1}{8n\sqrt{3}}\exp\left(\frac{2\pi}{\sqrt{6}}n^{1/2}\right)(1+o(1)). \end{equation*} Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph $\Gamma_2(A_n)$.