Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The product of two elements from $\Lambda_n$ can always be expressed as a linear combination of Schur polynomials, for example ($n=3$):
$$ (\boldsymbol{s}_{2,1,1}+2\boldsymbol{s}_{3,2,1})(\boldsymbol{s}_{2,0,0}+\boldsymbol{s}_{1,1,1})= 2\,\boldsymbol{s}_{{5,2,1}}+2\,\boldsymbol{s}_{{4,3,2}}+2\,\boldsymbol{s}_{{4,3,1} }+2\,\boldsymbol{s}_{{4,2,2}}+\boldsymbol{s}_{{4,1,1}}+2\,\boldsymbol{s}_{{3,3,2}} +\boldsymbol{s}_{{3,2,2}}+\boldsymbol{s}_{{3,2,1}} $$
On the other hand, $$ \boldsymbol{s}_{{3,0,0}}-\,\boldsymbol{s}_{{2,1,0}} +\boldsymbol{s}_{{1,1,1}} $$ cannot be expressed as a product of two non trivial elements from $\Lambda_n$.
Question: Consider an arbitrary element of $\Lambda_n$: $$ A= \sum_{\lambda} c_\lambda \boldsymbol{s}_{\lambda}, c_\lambda \in \mathbb{Q}, $$ where the sum is taken over a certain set of partitions $\lambda$.
Is there any criterion, in terms of the coefficients $c_\lambda$, that $A$ be the product of two symmetric polynomials?
Of course, we can express $A$ as polynomials in terms of, for example, the elementary symmetric polynomials $\boldsymbol{e}_i$ and then try to factorize it in the ring of polynomials, but I am interested in an intrinsic criterion, in terms of the coefficients $c_\lambda$.
Thank you.
