For a compact (reductive/semisimple) Lie group $G$ with a maximal torus $T$, which I will identify with a subgroup of ${\mathbb{C}^*}^n$ (for simplicity let's just say that $G\leqslant GL(n,\mathbb{C})$ and $T$ is its subgroup of diagonal matrices). Then one can take a character $\alpha$ of $T$: $$ \alpha \big( \operatorname{diag}(\lambda_1,\ldots,\lambda_n) \big) = \lambda_1^{p_1}\cdot\ldots\cdot\lambda_n^{p_n}$$ and, provided some restrictions (inequalities) on $p_1,\ldots,p_n$ are satisfied, extend it continuously to a function $\alpha(g)$ on a dense subset of $G$ such that all the shifts $\alpha(gg_0)$, $g_0\in G$, span a finite-dimensional subspace $\Phi_\alpha$ of $L^2(G)$. Then this span is an invariant subspace under the right regular representation and is the representation space of an irreducible representation $\pi_\alpha$. Moreover, all finite-dimensional irreducible representations of $G$ can be obtained this way.
Now if $\alpha$ and $\beta$ are two such characters of $T$, then so is $\alpha\beta$, and it so happens that the linear span of all products $f_\alpha f_\beta$, where $f_\alpha\in\Phi_\alpha$, $f_\beta\in\Phi_\beta$, equals $\Phi_{\alpha\beta}$. The representation $\pi_{\alpha\beta}$ is called the Young product of $\pi_\alpha$ and $\pi_\beta$. This is particularly useful when presenting an arbitrary irrep as a product of the fundamental representations, which is very simple to do based on the signature $(p_1,\ldots,p_n)$.
Q. What I wonder is whether there exists a formula which allows to express the character of $\pi_{\alpha\beta}$ in terms of the characters of $\pi_\alpha$ and $\pi_\beta$ (and possibly some other characters with smaller signatures).
Such a formula would not be very simple, I imagine, because while $\pi_{\alpha\beta}$ occurs in $\pi_\alpha \otimes \pi_\beta$ with multiplicity 1, there are many other summands in its decomposition into the irreducibles. On the other hand, there are simple formulas for the characters for the exterior and symmetric squares of a representation...
One reason to believe that there is some sort of explicit relation here is that for the fundamental representations the characters can be expressed as some simple symmetric polynomials in $\lambda_1,\ldots,\lambda_n$ (and their inverses), while the character of a representation, restricted onto $T$, is again a symmetric polynomial. For example, in $SU(n)$ any irreducible character can be expressed as an explicit determinant of a matrix of complete homogeneous symmetric polynomials in the parameters of the torus. In a similar vein, there is also Kostant's reformulation of Weyl character formula, which provides another denominator-free expression for the character. But somehow it seems that this questions is avoided in the literature (even when the characters of the fundamental representations and the Young products are discussed in the same section of the text). But the literature is quite extensive, so I might have easily missed it.
