Is there a formula for the character of $V\otimes_{Sym}W$ in terms of the two characters $X_V$ and $X_W$ ?
I am trying to find the general formula for the character of $Sym^n\ V$ and I don't know how to approach this problem.
We know that $X_{Sym^2\ V} (g)= \frac{1}{2}(X_V(g)^2+ (X_V(g^2))$ , and that
$V\otimes_{Sym}W$ has as eigenvalues the set $\{\lambda_i\mu_j|\ i\leq j\}_{i,j=1}^{n,\ m}$ where $n$ is the smallest of the dimensions of $V$ and $W$ and $m$ is the greatest, and the $\{\lambda_i\}$'s and $\{\mu_j\}$'s are respectively the eigenvalues of the space of the smallest dimension (say $V$) and the greatest (say $W$) , we can write
$X_V(g).X_W(g)= (\sum \lambda_i)(\sum \mu_j)= \sum_{i\leq j} \lambda_i\mu_j + \sum_{i> j} \lambda_i\mu_j= X_{V\otimes^{Sym} W}\ (g)+ X_{V\otimes^{Alt}\ W}\ (g)- \sum_{i<j} \lambda_i\mu_j$