Group representations that resist random sampling. Issue 3 (9th June 2014)

• Record Type: Journal Article
• Authors: Lovett, Shachar; Moore, Cristopher; Russell, Alexander
• Is Part Of: Random structures & algorithms. Volume 47:Issue 3(2015)
• Page Start: 605
• Page End: 614
• Publication Date: 2014-06-09
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20555
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We show that there exists a family of groups <italic>G</italic><sub><italic>n</italic></sub> and nontrivial irreducible representations ρ<sub><italic>n</italic></sub> such that, for any constant <italic>t</italic>, the average of ρ<sub><italic>n</italic></sub> over <italic>t</italic> uniformly random elements <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9tvbmw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20555:rsa20555-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>, </mml:mo><mml:mo>…</mml:mo><mml:mo>, </mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> has operator norm 1 with probability approaching 1 as <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9tvbkb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20555:rsa20555-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. More quantitatively, we show that there exist families of finite groups for which <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9tvbqj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20555:rsa20555-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Ω</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mi>log</mml:mi><mml:mo>|</mml:mo><mml:mi>G</mml:mi><mml:mo>|</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> random elements are required to bound the norm of a typical representation below 1. This settles a conjecture of A. Wigderson. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 605–614, 2015</p> </abstract>