For quantum affine A_n, the dominant ℓ-weights of every Weyl module are exactly the interval products in an explicit finite closure, Hom spaces between Weyl modules are at most one-dimensional, and the socle is described.
Alternating snake modules and a determinantal formula
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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 $\pm 1$.
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On dominant $\ell$--weights and maps between Weyl modules for quantum affine $A_n$
For quantum affine A_n, the dominant ℓ-weights of every Weyl module are exactly the interval products in an explicit finite closure, Hom spaces between Weyl modules are at most one-dimensional, and the socle is described.