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Character values for a finite group are sums of nth roots of unity. I'm wondering if there are any results bounding nonzero values of irreducible characters away from zero. Or if not are there examples of natural sequences of groups where entries in the character table can be made arbitrarily close zero. I don't see anything like this in Isaacs book for example.

Added: Thanks for the answers below, I of course should have figured out the direct product once you have one of norm <1. I guess the problem is more interesting for simple groups, for example the alternating groups the nonzero values are bounded away from zero.

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    $\begingroup$ Well, once you find a sum of roots of unity whose absolute value is less than $1,$ but non-zero, (which isn't difficult) you can realise that as the value of an irreducible character of a wreath product of the form $C_{a} \wr C_{b}.$ Then, taking direct products of such groups, you can get an irreducible character value which has as small a non-zero absolute value as you like. $\endgroup$
    – Geoff Robinson
    Commented Jan 23, 2014 at 22:32
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    $\begingroup$ I think the two dimensional representations of the dihedral groups will already give such an example. $\endgroup$
    – Lucia
    Commented Jan 23, 2014 at 22:35
  • $\begingroup$ What do you mean by natural sequence of groups? In my opinion the problem in the present form has no meaning. But one can ask the following question: Let $\mathfrak{X}$ be a class of finite groups and $\varepsilon$ be a positive number. Is there any group $G\in \mathfrak{X}$, a character $\chi\in Irr(G)$ and an element $g\in G$ such that $0<|\chi(g)|\leq \varepsilon$? $\endgroup$
    – Sh.M1972
    Commented Jan 24, 2014 at 4:17
  • $\begingroup$ The number of nonzero entries in the character table of the symmetric group, Sn, is among Sloane's sequences OEIS: OEIS A061256 #1st = number of ordered commuting 3-tuples in S[n] divided by n! n = 1 2 3 4 5 6 7 8 9 10 11 12 13 1,4,8,21,39,92,170,360,667,1316,2393,4541,8100 = = = == == == === xxx 1,4,8,21,39,92,170,331,593,1176,2118,3699,6658,11347 OEIS A006908 #2nd = number of non-zero entries in the character table of S[n]. ----------------- Differences 29 74 140 275 842 1442 $\endgroup$
    – john mckay
    Commented Feb 26, 2014 at 11:58
  • $\begingroup$ For small sums of roots of unity, see mathoverflow.net/questions/46068/… Maybe the question here should be, what kind of lower bound can you find in terms of the order of the group and/or the degree of the representation? $\endgroup$
    – Gerry Myerson
    Commented Feb 26, 2014 at 22:18

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Although as has been observed, $|\chi(x)|$ can be positive but arbitrarily close to zero, I think it is interesting that whenever $\chi(x)$ is not a root of unity or zero, then $|\chi(y)| > 1$ for some element $y$ that generates the same cyclic subgroup as $x$.

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The dihedral group $D_{2n}$ has two dimensional irreducible representations for which the character values (on the cyclic group of order $n$) are $2\cos(2\pi k/n)$ for $1\le k\le n$. Clearly these values get arbitrarily close to zero for large $n$ and $k$ close to $n/4$. See for example http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_dihedral_groups .

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  • $\begingroup$ "Clearly these values get arbitrarily close to zero for large $n$" !!!??? $\endgroup$
    – Sh.M1972
    Commented Jan 24, 2014 at 3:53
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    $\begingroup$ @M.Shahryari: I mean of course that there are values $k$ (e.g. when $k$ is about $n/4$) where the value is close to $\cos(\pi/2)=0$. $\endgroup$
    – Lucia
    Commented Jan 24, 2014 at 4:02

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