While working on a paper with some colleagues (as part of the same project which motivated one of my questions on MathOverflow), I was led to a question that is not directly relevant, but seemed interesting, and which I don’t know how to go about answering. I thought it might be salutary to write it up as a blog post and see if people have suggestions.

The question seems slightly too open-ended to be asked on MathOverflow, and more importantly my hope is for co-operative discussion, which is not really what I think MO is suited for. Maybe it has already been answered in the literature, or is somehow known to be as hard as problems that are still open.

Alors.

The problem concerns the irreducible characters of finite groups (over the complex field, in case that makes a difference), and how big the values of such characters can be on elements of a given group. To avoid needless repetition and tiresome caveats: all my groups in this post are finite and have at least two elements.

If G is a group and $\psi$ is an irreducible character on G, then $\vert\psi(x)\vert = \psi(e)$ for every $x\in Z(G)$. (Writing $d$ for $\psi(e)$, just take an irrep whose trace is $\psi$ and note that it has to send $x$ to a scalar multiple of the $d\times d$ identity matrix.) In general, though, $\vert\psi(x)\vert$ can be much smaller than $\psi(e)$.

Disregarding the sage words of Pedersen, let’s say that the MCR of an irreducible character $\psi$ is the maximum value of $\psi(e)^{-1}\vert\psi(x)\vert$ as $x$ runs over all non-identity elements of the group in question. Here are some of the basic properties of the MCR:

• The MCR of any irreducible character is strictly positive and at most 1.
• Any character that comes from a 1-dimensional representation (a.k.a. a “linear character”) clearly has MCR equal to 1.
• By the previous remarks, if G has non-trivial centre, then every irreducible character on G has MCR equal to 1.
• If G has trivial centre, there exists at least one irreducible character on G whose MCR is strictly less than 1.

(The last of these facts is a paraphrase, for those who know of these things, of the result that the intersection of all centres of all irreducible characters is the centre of the group.)

Here is my question:

Does there exists a constant $\alpha \in(0,1)$ such that each group with trivial centre possesses at least one irreducible character whose MCR is $\leq \alpha$?

Or, for those who prefer formulas to wordy formulations:

Is $\sup_{G: |Z(G)|=1} \min_{\psi\in\mathop{\rm Irr}(G)} \mathop{\rm MCR}(\psi) < 1$?

Some disjointed and simple-minded thoughts:

• My limited understanding of the literature is that people have mostly focused on getting upper bounds for the MCR that apply to all non-linear characters on particular classes of simple group: there is work of Gluck in this direction for groups of Lie type. However, this in some sense has different quantifiers from my question, and I have some vague hope that my question might be susceptible to global averaging/probabilistic arguments that don’t need as much detailed knowledge of the structure of finite groups…
• The simple-minded estimate $|x^G| |\psi(x)|^2 < \sum_{y\in G} |\psi(y)|^2 = |G|$, where $|x^G|$ denotes the size of the conjugacy class of $x$ in G, suggests that finding upper bounds on the minimal MCR we should focus on non-identity elements in the group with large centralizers, but I don't know how far one can get with that line of thought.
• In the absence of any particular proof strategy, one could start by taking one’s favourite family of groups with trivial centre and known character table, and then getting an upper bound on the minimal MCR by inspection of the formulas. For instance, if $G = PSL(2,F)$ where F is a finite field with $q$ elements, the character tables tell us that G has an irreducible character* of degree q which has absolute value at most 1 on all non-identity elements of G; hence the MCR of this character is $1/q$.

[*] The character in question is the Steinberg character, obtainable as the non-trivial summand of the permutation representation given by the usual action of PSL(2,F) on the F-projective line; it is apparently Frightfully Important, for reasons which are too advanced for this Bear of Little Brain.