The theory of $q$-characters for quantum affine algebras are studied in The q-characters of representations of quantum affine algebras and deformations of W-algebras, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, and Spectra of Tensor Products of Finite Dimensional Representations of Yangians.
I think that the following results are true. But I didn't find their proofs in the above references. Do the results follow from the general theory of $q$-characters? Thank you very much.
Let $m_1, \ldots, m_k$ be dominant monomials in $\chi_q(M_1) \chi_q(M_2)$, where $M_1, M_2$ are two simple $U_q(\hat{g})$-modules and $\chi_q(M)$ is the $q$-character of $M$. Then we have the following two results.
Suppose that $m_i$ is not contained in any $\chi_q(L(m_j))$, $j \neq i$. Then $L(m_i)$ is an irreducible subfactor of $M_1 \otimes M_2$. Here $L(m_i)$ is the simple $U_q(\hat{g})$-module with highest $l$-weight $m_i$.
Suppose that $L(m_i)$ is a subfactor of $M_1 \otimes M_2$. Suppose that $m_j$ appears in $\chi_q(M_1) \chi_q(M_2)$ with multiplicity $p$ and $m_j$ appears in $\chi_q(L(m_i))$ with multiplicity $p$. Then $m_j$ is not a subfactor of $M_1 \otimes M_2$.
