# The density of representation degrees

Martin Liebeck; Dan Segal; Aner Shalev

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 5, page 1519-1537
- ISSN: 1435-9855

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topLiebeck, Martin, Segal, Dan, and Shalev, Aner. "The density of representation degrees." Journal of the European Mathematical Society 014.5 (2012): 1519-1537. <http://eudml.org/doc/277618>.

@article{Liebeck2012,

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

author = {Liebeck, Martin, Segal, Dan, Shalev, Aner},

journal = {Journal of the European Mathematical Society},

keywords = {dimensions of irreducible complex representations; representation degrees; complex simple algebraic groups; arithmetic groups; finitely generated linear groups; asymptotics of density functions; virtually Abelian groups; dimensions of irreducible complex representations; representation degrees; complex simple algebraic groups; arithmetic groups; finitely generated linear groups; asymptotics of density functions; virtually Abelian groups},

language = {eng},

number = {5},

pages = {1519-1537},

publisher = {European Mathematical Society Publishing House},

title = {The density of representation degrees},

url = {http://eudml.org/doc/277618},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Liebeck, Martin

AU - Segal, Dan

AU - Shalev, Aner

TI - The density of representation degrees

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 5

SP - 1519

EP - 1537

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

LA - eng

KW - dimensions of irreducible complex representations; representation degrees; complex simple algebraic groups; arithmetic groups; finitely generated linear groups; asymptotics of density functions; virtually Abelian groups; dimensions of irreducible complex representations; representation degrees; complex simple algebraic groups; arithmetic groups; finitely generated linear groups; asymptotics of density functions; virtually Abelian groups

UR - http://eudml.org/doc/277618

ER -

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