I previously mentioned that I once made (in a footnote) the false claim that for a 11-dimensional representation V of the Mathieu group M_12, the 120 dimensional representation Ad^0(V) was irreducible. I had wanted to write down representations W of large dimension n such that Ad^0(W) of dimension n^2 – 1 was irreducible. In the comments, Emmanuel Kowalski pointed to a paper of Katz where he discusses actual examples (including the 1333 dimensional representation of the Janko group J_4). On the other hand, I recently learned from Liubomir Chirac’s thesis:

https://thesis.library.caltech.edu/8942/1/Chiriac_Thesis.pdf

that it’s an open problem to determine whether there exists such a representation for all n (although he does write down infinitely many examples in prime power dimension). Chirac’s thesis also lead me to the paper of Magaard, Malle, and Tiep, who do classify all such examples for (central extensions of) simple groups. Turns out that I *could* have used M_12 after all, or rather the 10-dimensional representation of the double cover 2.M_12, which *does* have the required property (the 99-dimensional representation factors through M_12, naturally).

One reason (amongst many) that (either of the) 11-dimensional representations V of M_12 do not have Ad^0(V) irreducible is that they are self-dual (oops). On the other hand, if you eyeball the character table, you will find that there *is* an irreducible representation W of dimension 120. Moreover, let me write down the characters of [V \otimes V^*] – [1] and [W]:

These seem surprisingly close to me! So now the question is, as one ranges over (some class perhaps all) finite groups G, what is the minimum number of conjugacy classes for which

\chi = [V \otimes V^*] – [1] – [W]

can be non-zero for irreducible V and W, assuming that it is non-zero? Since V is irreducible, by Schur’s Lemma, this virtual representation is orthogonal to [1] (unless [W] = [1] which would be silly). So which certainly implies that there must be at least *two* non-zero entries of opposite signs. I don’t see any immediate soft argument which pushes that bound to 3. I admit, this is a slightly silly question. But still, a beer to anyone who proves the example above is either optimal or comes up with an example with only two non-zero terms. (To avoid silliness, say that the dimension of V has to be at least 5.) The characters above look strikingly similar to me, and it does make we wonder if there is any reason for why they are so close. Perhaps if I knew more about groups, I could feel more confident in just chalking up the resemblance above to a law of small numbers.

Probably a more sensible question is to ask for how small the number of non-zero entries of of [V]-[W] can be for two distinct irreducibles. That question has surely been studied!

How about, G = the alternating group on 5 letters, V = the 5-dimensional representation, and W = the 4-dimensional representation?

This example is certainly at the borderline of silliness (I was imagining that \chi_{V \otimes V^*}(1) = \chi_W(1) + 1), but I stand by my beer offer. To get even more silly, one can let V = 5-dimensional representation of S_5, and W the zero representation!

There are a couple of bigger examples at the same borderline for GL_3(F_2). But don’t get too silly, zero’s not irreducible.