For $G$ a finite group and two $\mathbb CG$ modules $V$ and $W$, then (for $\chi_V$ the character ie the trace of the $G$-action) we have that
$$
\chi_{V \otimes W} = \chi_V \cdot \chi_W
$$
is a representation. I'm wondering in what conditions we can somehow reverse this, for instance taking two characters $\chi_V$ and $\chi_W$ and divide them to get a character $\phi := {\chi_V}/{\chi_W}$.
Edit: After the discussion in the comments, a more succinct formulation of my question is: given representations $\chi_V$ and $\chi_W$, what are conditions on existence of a third representation $U$ such that $V \cong W \otimes U$, so that
$$
\chi_V = \phi \cdot \chi_W
$$
with $\phi = \chi_U$ a character as opposed to a more general class function?
I realise necessary conditions would be that all the pointwise divisions would need to be algebraic integers, so for instance looking at the character table of $PSU_3(\mathbb F_2)$, $\rho_6/\rho_5$ won't be a character. So being finite solvable isn't sufficient (link indicating this group is $C_3^2 \rtimes Q_8$), nor is the dimension of one dividing the dimension of the other.
Equally on the same page, the character table of $C_3 \rtimes S_3$ shows $\dim V = \dim W$ isn't sufficient either…
I also see why this might not hold at all in generality, since my intuition specifically comes from small groups with characters which can be constructed by the tensor representation.
On the other hand, for one dimensional representations the characters are exactly homomorphisms into $S^1 \subseteq \mathbb C^\times$, and so we'll have $\chi_V /\chi_W = \chi_V \cdot \overline{\chi_W} = \chi_V \cdot \chi_{W^*}$ for $W^*$ the dual. So for instance in an abelian group, this is always true.
Are there known necessary/sufficient conditions for this to hold? Or is this an idea that simply doesn't generalise usefully?
Best Answer
Rather than talking about division of characters, you should formulate this in terms of factors (divisors if you will) of characters. Using that point of view, there exists an important theory for finite solvable groups that contains so-called factored characters: let $\pi$ be a non-empty set of primes. A finite group is called a $\pi$-group if the prime factors of its order lie in $\pi$, or equivalently, its order is a $\pi$-number. Write $\pi$' for the complement of $\pi$ in the set of prime numbers. A group is called $\pi$-separable if it has subnormal series $1=S_0 \subseteq S_1 \subseteq \cdots \subseteq S_r=G$ with each of the factor groups $S_i/S_{i-1}$ being either a $\pi$-group or a $\pi$'-group. It is easy to show that a group is solvable if and only if it is $\pi$-separable for all sets $\pi$. Now, $\pi$-separable groups on its own are worth to study: to a large extent they generalize Sylow theory to start with (replacing a Sylow $p$-subgroup by a Hall $\pi$-subgroup: e.g. existence and conjugation are guaranteed in $\pi$-separable groups).
Dilip Gajendragadkar, a student of Everett Dade, defined in 1978 in his PhD thesis a special class of characters with a rather technical definition (cleverly appealing to the abundance of (sub)normal subgroups being present in solvable and $\pi$-separable groups):
Definition Let $G$ be a $\pi$-separable group, then $\chi \in Irr(G)$ is called $\pi$-special if $\chi(1)$ is a $\pi$-number, and the determinantal order $o(\theta)$ is a $\pi$-number for every irreducible constituent $\theta$ of the restriction $\chi_S$, for all subnormal subgroups $S$ of $G$.
He then proved a remarkable theorem.
Theorem (D. Gajendragadkar, 1978) Let $G$ be a $\pi$-separable group and suppose that $\varphi, \psi \in Irr(G)$ are $\pi$-special and $\pi$'-special respectively. Then their product $\chi=\varphi\psi \in Irr(G)$. In addition, such a factorization in a $\pi$- and $\pi$'-special character is unique.
This result motivated researchers to look more deeply into $\pi$-factored characters as $\chi$ is in the theorem above. This led to the definition of fully factored characters: let $G$ be solvable, then $\chi \in Irr(G)$ is fully factored if it can be written as $\chi=\Pi \alpha_p$, where $p$ runs over some set $\pi$ of primes and each factor $\alpha_p$ is $\{p\}$-special. It turns out that primitive irreducible characters of solvable groups are fully factored. This ingredient led to a totally different proof of the famous Feit Conjecture on character values in cyclotomic fields for solvable groups (initially proven in 1986 by Amit & Chillag and independently by Ferguson & Turull). See also here for further information.