52
$\begingroup$

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,x_n)=g \}|. $$ The question is:

Can we characterize the words $w$ for which $f_w$ is a virtual character for any group $G$?

(A virtual character is a difference of two characters, sometimes also called generalized character.) That $f_w$ is a virtual character of $G$ is equivalent to
$$ (f_w,\chi) = \frac{1}{|G|} \sum_{x_1, \dots,x_n\in G} \chi(w(x_1, \dots,x_n)) \in \mathbb{Z} $$ for all $\chi \in \operatorname{Irr}(G)$.
Some examples and non-examples:

  1. $n=1$, $w= x^k$. Then $(f_w,\chi)\in \mathbb{Z}$ is the $k$-th Frobenius-Schur-indicator.

  2. $w=[x,y]=x^{-1}y^{-1}xy$. Then $(f_w,\chi)= \frac{|G|}{\chi(1)}$, so $f_w$ is in fact a character.

  3. For $w=x^{-k}y^{-1}x^ly$, it can be shown that $(f_w,\chi)= \frac{|G|}{\chi(1)} a$ for some integer $a$.

  4. $f_w$ is a virtual character for words of the form $w= x_1^{e_1}x_2^{e_2} \dots x_n^{e_n}$.

  5. For $w=[x^2,y^2]$ and $G= SL(2,7)$, the class function $f_w$ is not a generalized character.

It is always the case that $(f_w, \chi)$ is an algebraic integer in $\mathbb{Q}(\chi)$. This follows from an old result of Solomon saying $f_w(g)$ is divisible by $|C_G(g)|$ when the number of variables is $>1$. (See also this paper by Rodriguez Villegas and Gordon, where my question is discussed very briefly.) So for rational-valued groups, $f_w$ is a virtual character for any word. (In fact, it suffices that $G$ is a normal subgroup of a rational-valued group.) In view of 4., the same is true for abelian groups, and I don't know examples of solvable groups where $f_w$ is not a virtual character.
Another remark is that $f_w$ does not change if we replace $w$ by its image under an automorphism of the free group in question. Using this, one can of course find rather complicated looking words for which $f_w$ is trivial, that is, $f_w(g)\equiv |G|^{n-1}$.

Added later:
The proofs that $(f_w,\chi)\in \mathbb{Z}$ in 2.-4. above basically use the same "trick". I'm also interested in other examples where perhaps another idea of proof is used than in the examples 1.-4., or "more conceptual" explanations why $f_w$ sometimes is a (virtual) character. (I admit that this is somewhat vague.)

A related question is when $f_w$ is a character, not only a virtual character. For $G$ the symmetric group, this is discussed in Exercise 7.69k of "Enumerative Combinatorics 2" by Richard Stanley. (Note that in this case, $f_w$ is always a virtual character.)

$\endgroup$
2
  • 1
    $\begingroup$ This question is considered (although I don't think answered definitively) in this papr of Amit and Vishne: u.math.biu.ac.il/~vishne/publications/S0219498811004690.pdf . $\endgroup$
    – HJRW
    Sep 3, 2020 at 10:23
  • $\begingroup$ To add to HJRW's comment a function is a $\mathbb Q$-linear combination of characters iff it is constant on elements generating conjugate subgroups. In your set up because it is an algebraic integer combination of characters you just need it to be a $\mathbb Q$-linear combination to be a virtual character. So at the end of the day you need that if $m$ is relatively prime to the order of $G$ then the number of solutions for $g$ and $g^m$ are the same. For the symmetric group this happens always because $g$ and $g^m$ are conjugate. The same is true for any group with all characters rational. $\endgroup$ Oct 3, 2021 at 13:15

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.