Alternating snake modules and a determinantal formula

TL;DR

This paper introduces a new family of modules for quantum affine algebras, providing criteria for primality, a determinantal formula, and applications to classical representation theory questions.

Contribution

It defines a novel family of modules encompassing snake modules and cluster algebra modules, with prime conditions, a determinantal formula, and applications to category O.

Findings

Modules include snake and cluster algebra modules

Prime modules characterized by necessary and sufficient conditions

Explicit alternating sum formula for modules

Abstract

We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the 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 classical questions 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 $\pm 1$.