Application of character estimates to the number of 𝑇₂-systems of the alternating group

Abstract

Abstract We use character theory and character estimates to show that the number of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>T</m:mi><m:mn>2</m:mn></m:msub></m:math> {T_{2}} -systems of the alternating group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub></m:math> {A_{n}} is at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mfrac><m:mn>1</m:mn><m:mrow><m:mn>8</m:mn><m:mo>⁢</m:mo><m:mi>n</m:mi><m:mo>⁢</m:mo><m:msqrt><m:mn>3</m:mn></m:msqrt></m:mrow></m:mfrac><m:mo>⁢</m:mo><m:mrow><m:mi>exp</m:mi><m:mo>⁡</m:mo><m:mrow><m:mo maxsize="210%" minsize="210%">(</m:mo><m:mrow><m:mfrac><m:mrow><m:mn>2</m:mn><m:mo>⁢</m:mo><m:mi>π</m:mi></m:mrow><m:msqrt><m:mn>6</m:mn></m:msqrt></m:mfrac><m:mo>⁢</m:mo><m:msup><m:mi>n</m:mi><m:mrow><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msup></m:mrow><m:mo maxsize="210%" minsize="210%">)</m:mo></m:mrow></m:mrow><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:mrow><m:mi>o</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>1</m:mn><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo>.</m:mo></m:mrow></m:math> \frac{1}{8n\sqrt{3}}\exp\biggl{(}\frac{2\pi}{\sqrt{6}}n^{1/2}\biggr{)}(1+o(1)). Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi mathvariant="normal">Γ</m:mi><m:mn>2</m:mn></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:msub><m:mi>A</m:mi><m:mi>n</m:mi></m:msub><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\Gamma_{2}(A_{n})} .

Locations

• Journal of Group Theory - View - PDF