Combinatorial Identities Related to Representations of $U_q(\widetilde{\mathfrak{gl}_2})$

We present certain combinatorial identities related to tensor products of evaluation representations of the quantum loop algebra $U_q(\widetilde{\mathfrak{gl}_2})$ or the elliptic quantum group $E_{\rho,\gamma}(\mathfrak{sl}_2)$. The simplest example of the obtained identities was discovered by N. Jing from the validity of the Serre relations in some vertex representations of quantum Kac–Moody algebras.