Alternating Snake Modules and a Determinantal Formula

We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to a classical question in the category \(\mathcal O(\mathfrak {gl}_r)\). Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights \(\mu \) for which the non-zero Kazhdan–Lusztig coefficients \(c_{\mu , \nu }\) are \(\pm 1\).