[seqfan] Smallest groups with an irreducible representation of given dimension
David Harden
sylow2subgroup at gmail.com
Sat Apr 6 11:36:35 CEST 2013
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
representation).
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
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