A path description for $\epsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity

Abstract

Fix ${\epsilon }^{2\ell }=1$ with $\ell \ge 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_{n+1})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\epsilon$-characters of any simple module with highest $l$-weight of degree two and any Kirillov-Reshetikhin module of $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_{n+1})$. As an application of our path description, we obtain a necessary and sufficient condition for the tensor product of two fundamental representations of $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_{n+1})$ to be irreducible. Additionally, we obtain a necessary condition for the tensor product of two or more fundamental representations of $U_{\epsilon }^{\mathrm{res}}(L{\mathfrak{sl}}_{n+1})$ to be irreducible.

References in corpus (5)

• Cluster algebras and quantum affine algebras
• Path description of type B q-characters
• Simple tensor products
• Extended T-systems
• Path description for $q$-characters of fundamental modules in type $C$