Why would dim primitive irrep divide size of some conjugacy class ?

Question

From Isaacs et.al. 2005

Conjecture C. Let χ be a primitive irreducible character of an arbitrary finite group G. Then χ(1) divides | clG(g)| for some element g ∈ G.

Here, of course, we have written clG(g) to denote the class of g in G. We have checked that Conjecture C holds for all irreducible characters (primitive or not) of all groups in the Atlas 1.

Question 1 What is motivation for this ? Is it possible to describe what conjugacy class(es) should correspond to irreducible representation in this way ? (at least for some standard groups S_n, A_n, GL_n(F_q),...) What are representative examples?

Question 2 Is it still open ?

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The authors write:

We now digress to explain our original motivation for considering these questions. There are numerous parallels and analogies between theorems concerning the of set irreducible character degrees of a finite group and theorems concerning the set of conjugacy class sizes of such groups. This suggests that perhaps there are some subtle arithmetic connections between these two sets of integers associated with a given group. One such connection that is easy to see is that each prime number that divides an irreducible character degree of G must also divide some class size of G. If G is solvable, then S. Dolfi showed that more is true. He proved [2] that given any two distinct primes p and q such that pq divides some irreducible character degree of a solvable group G, then pq also divides some class size of G. One might conjecture that the analogous assertion for three or more distinct primes is also true, but as far as we know, this remains open.

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Partial result:

In the following, we use the notation np to denote the p-part of a positive integer n, where p is a prime number.

Corollary D. Let χ be a primitive irreducible character of a solvable group G, and let p be a prime divisor of |G|. Then χ(1)p divides (| clG(g)|p) 3 for some element g ∈ G.

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Not related results, for complteness:

Denote CV(g) fixed point subspace for g in V.

Our main result is the following.

Theorem A. Let V be a nonzero finite dimensional completely reducible F G-module, where F is any field and G is any finite group. Assume that CV (G) = 0 and let p be the smallest prime divisor of |G|. Then there exists some element g ∈ G such that

$ dim CV (g) ≤ (1/p) ~ dim V $.

The fraction 1/p cannot, in general, be replaced by any smaller quantity. In particular, this shows that Neumann’s conjecture is valid for odd-order groups, at least...

Corollary B. Let V be a nonzero finite dimensional completely reducible F G-module, where F is an arbitrary field and G is any finite group, and assume that CV (G) = 0. Then

$1/ |G| \sum_{g∈G} dim CV (g) ≤ ((p + 1)/ 2p)~~ dim V$ ,

where p is the smallest prime divisor of |G|.

Answer 1

I have not noticed this question before, though it was posted several years ago. As a comment on the question as a whole, and especially Question 1 asked in the text, there are likely to be many such elements $g$ for many groups, and I would not expect there to be any "natural correspondence" between the $g$ associated in this way to a given $\chi.$

The reason I put forward that comment is that a result of J.G. Thompson (which appears in the text by Isaacs on character theory).

The result of Thompson asserts that if $\chi$ is an irreducible character (primitive or not) of finite group $G$ then $\chi$ takes value $0$ or a root of unity on more than $\frac{|G|}{3}$ elements of $G$.

This does not prove the conjecture, since it could be that $\chi$ vanishes on more than $\frac{|G|}{3}$ elements ( and extraspecial $p$-groups of order $p^{3}$- $p$ a prime - are examples where there is an irreducible character taking root of unity values nowhere, though the irreducible character is imprimitive in that case).

But it does make it seem likely that there will be many groups $G$ which have an irreducible character $\chi$ taking a root of unity value at some $x \in G,$ and often one might expect several such $x$.

For any such $x,$ note that $\frac{[G:C_{G}(x)]}{\chi(1)} = \overline{\chi(x)}\frac{[G:C_{G}(x)]\chi(x)}{\chi(1)} $ is a rational algebraic integer, hence is an integer.

Later edit: A familiar example is the Steinberg character $\chi$ of a finite quasisimple characteristic $p$ Lie type group $G$. For each $p$-regular $g \in G,$ we have $\chi(g) = \pm |C_{G}(g)|_{p},$ and by Brauer's general theory we have $p \not | |C_{G}(g)|$ for some $p$-regular $g \in G$, so that $\chi(1)$ (which is a power of $p$) divides $[G:C_{G}(g)].$

Later edit rewriting badly written earlier addition:

Thompson's argument uses a result of C. Siegel, stating that if $\alpha \neq 1$ is a totally positive (real) algebraic integer with $n$ algebraic conjugates, then the sum of those conjugates is at least $\frac{3n}{2}$ (the bound is attained for $\alpha = \frac{3 \pm \sqrt{5}}{2}),$ as Siegel noted.

We apply this to the orthogonality relations for the columns of the character table (instead of the rows, as Thompson did) to obtain an inequality relevant to this question.

Let $x$ be an element of the finite group $G$ such that $|C_{G}(x)| = c$ such that $s$ irreducible characters of $G$ do not vanish at $x$ and such that $a$ of those characters do not take root of unity values at $x$ either.

Then Siegel's result yields that $(s-a) + \frac{3a}{2} \leq c,$ so that $s \leq c- \frac{a}{2}.$ Note also that $s \geq a + [G:G^{\prime}],$ since every linear character of $G$ takes a root of unity value at $x.$

Hence $c \geq \frac{3a}{2} + [G:G^{\prime}]$ and $a \leq \frac{2(c-[G:G^{\prime}])}{3}.$

If there is no non-linear irreducible character of $G$ taking a root of unity value at $x,$ then we have $s = a +[G:G^{\prime}],$ so that $s \leq \frac{[G:G^{\prime}]}{3} + \frac{2c}{3}.$

We may conclude that if more than $\frac{[G:G^{\prime}]}{3} + \frac{2|C_{G}(x)|}{3}$ irreducible characters of $G$ do not vanish at $x,$ then there is a non-linear irreducible character $\chi$ of $G$ such that $\chi(x)$ is a root of unity (so that $\chi(1)$ divides $[G:C_{G}(x)]).$