[seqfan] Re: Smallest groups with an irreducible representation of given dimension
njasloane at gmail.com
Mon Apr 8 14:01:51 CEST 2013
David, I don't know that we have any group theorists on the Seq Fan list.
So I asked a group theorist friend (not on this list) to comment on your
proposed sequence. Here is part of the reply:
1, 6=S3, 12=A4, 20=C5:C4, 55=C11:C5, 42=C7:C6, 56=(C2xC2xC2):C7,
72=C3xC3:(some regular subgroup of L2(3) on 8 points),
171=C19:C9, 110 = C11:C10, 253=C23:C11, 156=C13:C12, 351=(C3xC3xC3):C13, ...
If one has a small cyclic group of order p such that varphi(p) is a
multiple of n,
then take Cp:Cn. For other n it is reasonable to take elementary abelian
groups of order n+1 that admit a subgroup of their automorphism group
acting regularly on the n non-trivial characters.
Can you go ahead and submit the sequence (of course only including the
terms as far as you are certain of them)?
On Sat, Apr 6, 2013 at 5:36 AM, David Harden <sylow2subgroup at gmail.com>wrote:
> The sequence I am considering is the one for which a_n is the order of the
> smallest finite group(s) possessing an irreducible complex representation
> of dimension n.
> a_1 = 1, and n > 1 implies (writing a_n as the sum of the squares of the
> dimensions of the group's irreducible representations) a_n > n^2 -- since
> a_n has to be a multiple of n, this yields a_n >= n^2 + n.
> There is a certain upper bound for this sequence which may always equal the
> sequence: this is nq, where q is the smallest prime-power congruent to 1
> mod n.
> A group of order nq possessing an irreducible representation of dimension n
> is given by the group of all functions from the finite field F_q to itself
> of the form f(x) = ax + b, where b is an arbitrary element of F_q and a is
> an element of F_q such that a^n = 1. This group has n 1-dimensional
> representations and (q-1)/n irreducible representations of dimension n.
> To see what else might happen to outdo this upper bound, consider the close
> call which occurs when n=5: the upper bound produces 55, which indeed
> works. But the next multiple of 5 is also the order of a group with an
> irreducible 5-dimensional representation, since 60 is the order of the
> alternating group A_5, which has an irreducible representation of dimension
> 5 (the doubly transitive 6-point representation minus the trivial
> I know it's not easy to estimate the smallest prime congruent to 1 mod n,
> and replacing "prime" with "prime-power" shouldn't help much. Under what
> assumptions can it be proven that the upper bound always equals the
> sequence (if n > 1)?
> For the sake of completeness: the first few values of the sequence,
> assuming the upper bound is tight for n > 1: 1, 6, 12, 20, 55, 42, 56, 72,
> 171, 110, 253, 156, 351, 406, 240, 272, 1751, 342, 3629, 820, 903, 506,
> 1081, 600, 2525, 702, 2943, 812, 1711, 930, 992, 3104, 2211, 3502, 2485,
> 1332, 5513, 7258, 3081, 1640, 3403, 1806, 7439, 3916, 8145, 2162, 13301,
> 2352, 9653, 5050, 5253, 2756 (this is the n=52 term)
> ---- David Harden
> Seqfan Mailing list - http://list.seqfan.eu/
Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
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Email: njasloane at gmail.com
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