JournalsjemsVol. 14, No. 5pp. 1519–1537

The density of representation degrees

  • Martin W. Liebeck

    Imperial College, London, UK
  • Aner Shalev

    The Hebrew University of Jerusalem, Israel
  • Dan Segal

    University of Oxford, United Kingdom
The density of representation degrees cover
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Abstract

For a group GG and a positive real number xx, define dG(x)d_G(x) to be the number of integers less than xx which are dimensions of irreducible complex representations of GG. We study the asymptotics of dG(x)d_G(x) for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an "alternative" for finitely generated linear groups GG in characteristic zero, showing that either there exists α>0\alpha > 0 such that dG(x)>xαd_G(x)>x^\alpha for all large xx, or GG is virtually abelian (in which case dG(x)d_G(x) is bounded).

Cite this article

Martin W. Liebeck, Aner Shalev, Dan Segal, The density of representation degrees. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1519–1537

DOI 10.4171/JEMS/339