Groups with irreducible representations of the largest possible dimension - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:44:30Z http://mathoverflow.net/feeds/question/97628 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97628/groups-with-irreducible-representations-of-the-largest-possible-dimension Groups with irreducible representations of the largest possible dimension Will Sawin 2012-05-22T05:12:07Z 2012-05-22T07:59:52Z <p>I thought that the interesting question Gerry Myerson asked in the comments of <a href="http://mathoverflow.net/questions/97602/degrees-of-irreducible-characters-of-groups-of-order-48-closed" rel="nofollow">this question</a> deserved to be asked in a non-closed mathoverflow question.</p> <p>What can we say about groups of order \$n\$ with an irreducible representation of dimension \$d\$ such that \$(d+1)^2\geq n\$?</p> <p>To ask a concrete question, are there infinitely many such groups?</p> http://mathoverflow.net/questions/97628/groups-with-irreducible-representations-of-the-largest-possible-dimension/97634#97634 Answer by Gerry Myerson for Groups with irreducible representations of the largest possible dimension Gerry Myerson 2012-05-22T06:06:26Z 2012-05-22T06:06:26Z <p>There are at least a handful. The groups of order 1, 2, and 3, and \$S_3\$, and the group of the square, and as I noted at that earlier question, \$A_4\$, and one of the groups of order 20, and one of the groups of order 42. </p> http://mathoverflow.net/questions/97628/groups-with-irreducible-representations-of-the-largest-possible-dimension/97635#97635 Answer by Michael for Groups with irreducible representations of the largest possible dimension Michael 2012-05-22T06:10:17Z 2012-05-22T06:10:17Z <p>There are infinitely many groups like that, since any Frobenius group of order d(d+1) where d+1 is a prime power has an irreducible representation of degree d. </p> http://mathoverflow.net/questions/97628/groups-with-irreducible-representations-of-the-largest-possible-dimension/97641#97641 Answer by Mark Sapir for Groups with irreducible representations of the largest possible dimension Mark Sapir 2012-05-22T07:34:22Z 2012-05-22T07:59:52Z <p>Just to give correct references. Let \$d\$ be the degree of an irreducible character of a finite group \$G≠1\$. Then \$|G|=d(d+e)\$ for some \$e > 0 \$ (that is because \$d\$ divides \$|G|\$ and \$d^2 &lt; |G|\$). Therefore the condition \$(d+1)^2 > |G|\$ means \$d(d+e)=|G|\$ with \$e=1\$ or \$2\$. If \$e=1\$, then \$G\$ is a doubly transitive Frobenius group or of order 2 by Berkovich, Yakov Groups with few characters of small degrees. Israel J. Math. 110 (1999), 325–332. If \$e=2\$, then \$G\$ is a cyclic group of order 3 or non-Abelian group of order \$8\$ by Snyder, Noah Groups with a character of large degree. Proc. Amer. Math. Soc. 136 (2008), no. 6, 1893–1903. In general the order of \$G\$ is bounded by \$((2e)!)^2\$ by Snyder's paper. That estimate was greatly improved to \$O(e^6)\$ in Isaacs, I. M. Bounding the order of a group with a large character degree. J. Algebra 348 (2011), 264–275 (using the Classification of finite simple groups) and then to \$e^6-e^4\$ (if \$e\ge 2\$) by Durfee, Christina, Jensen, Sara A bound on the order of a group having a large character degree. J. Algebra 338 (2011), 197–206 (without the Classification). On the other hand, by Snyder's remark, a finite version of the Heisenberg group gives a low bound \$e^4-e^3\$ and \$O(e^4)\$ is conjecturally also the upper bound (see Isaacs' paper where this upper bound is proved in many cases). </p>